# Exponential and Logarithmic Functions

## Exponential and Logarithmic Equations

### Learning Objectives

1. Solve Exponential Equations. (IA 10.2.2)
2. Solve Logarithmic Equations. (IA 10.3.4)

### Objective 1: Solve Exponential Equations. (IA 10.2.2)

Equations that include an exponential expression  are called exponential equations. There are two types of exponential equations: those with the common base on each side, and those without a common base.

Type 1: Possible common base on each side: Use properties of exponents to rewrite each side with a common base. Use base-exponent property to set exponents equal to each other and solve for x.
Type 2: No possible common base: Use properties of exponents to rewrite each side in terms of one exponential expression. Take the log or ln of each side and use the power rule to bring down the power. Solve the remaining equation for x.

### Practice Makes Perfect

Solve. Find the exact answer and then approximate it to three decimal places.

### Objective 2: Solving Logarithmic Equations.  (IA 10.3.4)

There are two types of logarithmic equations: those with log terms on just one side of the equation or those with log terms on each side of the equation.  Since the domain of logarithmic functions is positive numbers only, make sure to check the solutions.
Type 1: Log terms on one side of the equation: Use properties of logs to rewrite a side with just one log term. Convert to exponential notation and solve for x.
If  then  .
Type 2: Log terms on both sides of equation: First, use log properties to rewrite each side in terms of a single log expression, if necessary. Use the one-to-one property of logarithmic equality to set arguments equal to one another. Solve the resulting equation for x.

### Practice Makes Perfect

Don’t forget to check your solutions.

In 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions.
Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.

### Using Like Bases to Solve Exponential Equations

The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers

 and
 where

 if and only if

In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown.
For example, consider the equation
 To solve for
 we use the division property of exponents to rewrite the right side so that both sides have the common base,
 Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for
:

### Rewriting Equations So All Powers Have the Same Base

Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.
For example, consider the equation
 We can rewrite both sides of this equation as a power of
 Then we apply the rules of exponents, along with the one-to-one property, to solve for

### Solving Exponential Equations Using Logarithms

Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since
 is equivalent to
 we may apply logarithms with the same base on both sides of an exponential equation.

### Equations Containing e

One common type of exponential equations are those with base
 This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base
 on either side, we can use the natural logarithm to solve it.

### Extraneous Solutions

Sometimes the methods used to solve an equation introduce an extraneous solution, which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.

### Using the Definition of a Logarithm to Solve Logarithmic Equations

We have already seen that every logarithmic equation
 is equivalent to the exponential equation
 We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.
For example, consider the equation
 To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for

### Using the One-to-One Property of Logarithms to Solve Logarithmic Equations

As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers

 and any positive real number
 where

For example,
So, if
 then we can solve for
 and we get
 To check, we can substitute
 into the original equation:
 In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown.
For example, consider the equation
 To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for

To check the result, substitute
 into

### Solving Applied Problems Using Exponential and Logarithmic Equations

In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.
One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life.  lists the half-life for several of the more common radioactive substances.
We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:
where
1.
 is the amount initially present
2.
 is the half-life of the substance
3.
 is the time period over which the substance is studied
4.
 is the amount of the substance present after time

### Key Equations

### Key Concepts

1. We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
2. When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See .
3. When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See , , and .
4. When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See .
5. We can solve exponential equations with base
 by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See  and .
6. After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See .
7. When given an equation of the form
 where
 is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation
 and solve for the unknown. See  and .
8. We can also use graphing to solve equations with the form
 We graph both equations
 and
 on the same coordinate plane and identify the solution as the x-value of the intersecting point. See .
9. When given an equation of the form
 where
 and
 are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation
 for the unknown. See .
10. Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, use like bases to solve the exponential equation.
For the following exercises, use logarithms to solve.
For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.
For the following exercises, use the definition of a logarithm to solve the equation.
For the following exercises, use the one-to-one property of logarithms to solve.
For the following exercises, solve each equation for

### Graphical

For the following exercises, solve the equation for
 if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.
For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

### Technology

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.
For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

### Extensions


# Exponential and Logarithmic Functions

## Exponential and Logarithmic Models

### Learning Objectives

1. Use exponential models in applications. (IA 10.2.3)
2. Use logarithmic models in applications. (IA 10.3.5)

### Objective 1: Use exponential models in applications. (IA 10.2.3)

### Using exponential models

Exponential functions model many situations. If you have a savings account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, P, is invested at an interest rate, r, for t years, the new balance, A, will depend on how often the interest is compounded.

### Exponential Growth and Decay

Other topics that are modeled by exponential functions involve growth and decay. Both also use the formula  we used for the growth of money. For growth and decay, generally we use  as the original amount instead of calling it  the principal. We see that exponential growth has a positive rate of growth and exponential decay has a negative rate of growth.

### Practice Makes Perfect

### Objective 2: Use logarithmic models in applications. (IA 10.3.5)

### Decibel Level of Sound

There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from 0, which is barely audible to 160, which can rupture an eardrum. The10-12 in the formula represents the intensity of sound that is barely audible.

The magnitude  of an earthquake is measured by a logarithmic scale called the Richter scale. The model is  where  is the intensity of the shock wave. This model provides a way to measure earthquake intensity.

### Practice Makes Perfect

Use logarithmic models in applications.
We have already explored some basic applications of exponential and logarithmic functions. In this section, we explore some important applications in more depth, including radioactive isotopes and Newton’s Law of Cooling.

### Modeling Exponential Growth and Decay

In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth, we may choose the exponential growth function:
where  is equal to the value at time zero,  is Euler’s constant, and  is a positive constant that determines the rate (percentage) of growth. We may use the exponential growth function in applications involving doubling time, the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time. In some applications, however, as we will see when we discuss the logistic equation, the logistic model sometimes fits the data better than the exponential model.
On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably choose the exponential decay model. Again, we have the form  where  is the starting value, and  is Euler’s constant. Now  is a negative constant that determines the rate of decay. We may use the exponential decay model when we are calculating half-life, or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes.
In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph. Exponential growth and decay graphs have a distinctive shape, as we can see in  and . It is important to remember that, although parts of each of the two graphs seem to lie on the x-axis, they are really a tiny distance above the x-axis.
Exponential growth and decay often involve very large or very small numbers. To describe these numbers, we often use orders of magnitude. The order of magnitude is the power of ten, when the number is expressed in scientific notation, with one digit to the left of the decimal. For example, the distance to the nearest star, Proxima Centauri, measured in kilometers, is 40,113,497,200,000 kilometers. Expressed in scientific notation, this is  So, we could describe this number as having order of magnitude

### Half-Life

We now turn to exponential decay. One of the common terms associated with exponential decay, as stated above, is half-life, the length of time it takes an exponentially decaying quantity to decrease to half its original amount. Every radioactive isotope has a half-life, and the process describing the exponential decay of an isotope is called radioactive decay.
To find the half-life of a function describing exponential decay, solve the following equation:
We find that the half-life depends only on the constant  and not on the starting quantity
The formula is derived as follows

Since  the time, is positive,  must, as expected, be negative. This gives us the half-life formula

### Radiocarbon Dating

The formula for radioactive decay is important in radiocarbon dating, which is used to calculate the approximate date a plant or animal died. Radiocarbon dating was discovered in 1949 by Willard Libby, who won a Nobel Prize for his discovery. It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the ratio of those two isotopes in the air. It is believed to be accurate to within about 1% error for plants or animals that died within the last 60,000 years.
Carbon-14 is a radioactive isotope of carbon that has a half-life of 5,730 years. It occurs in small quantities in the carbon dioxide in the air we breathe. Most of the carbon on Earth is carbon-12, which has an atomic weight of 12 and is not radioactive. Scientists have determined the ratio of carbon-14 to carbon-12 in the air for the last 60,000 years, using tree rings and other organic samples of known dates—although the ratio has changed slightly over the centuries.
As long as a plant or animal is alive, the ratio of the two isotopes of carbon in its body is close to the ratio in the atmosphere. When it dies, the carbon-14 in its body decays and is not replaced. By comparing the ratio of carbon-14 to carbon-12 in a decaying sample to the known ratio in the atmosphere, the date the plant or animal died can be approximated.
Since the half-life of carbon-14 is 5,730 years, the formula for the amount of carbon-14 remaining after  years is

where

This formula is derived as follows:
To find the age of an object, we solve this equation for
Out of necessity, we neglect here the many details that a scientist takes into consideration when doing carbon-14 dating, and we only look at the basic formula. The ratio of carbon-14 to carbon-12 in the atmosphere is approximately 0.0000000001%. Let  be the ratio of carbon-14 to carbon-12 in the organic artifact or fossil to be dated, determined by a method called liquid scintillation. From the equation  we know the ratio of the percentage of carbon-14 in the object we are dating to the initial amount of carbon-14 in the object when it was formed is  We solve this equation for  to get

### Calculating Doubling Time

For decaying quantities, we determined how long it took for half of a substance to decay. For growing quantities, we might want to find out how long it takes for a quantity to double. As we mentioned above, the time it takes for a quantity to double is called the doubling time.
Given the basic exponential growth equation
 doubling time can be found by solving for when the original quantity has doubled, that is, by solving

The formula is derived as follows:
Thus the doubling time is

### Using Newton’s Law of Cooling

Exponential decay can also be applied to temperature. When a hot object is left in surrounding air that is at a lower temperature, the object’s temperature will decrease exponentially, leveling off as it approaches the surrounding air temperature. On a graph of the temperature function, the leveling off will correspond to a horizontal asymptote at the temperature of the surrounding air. Unless the room temperature is zero, this will correspond to a vertical shift of the generic exponential decay function. This translation leads to Newton’s Law of Cooling, the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature
This formula is derived as follows:

### Using Logistic Growth Models

Exponential growth cannot continue forever. Exponential models, while they may be useful in the short term, tend to fall apart the longer they continue. Consider an aspiring writer who writes a single line on day one and plans to double the number of lines she writes each day for a month. By the end of the month, she must write over 17 billion lines, or one-half-billion pages. It is impractical, if not impossible, for anyone to write that much in such a short period of time. Eventually, an exponential model must begin to approach some limiting value, and then the growth is forced to slow. For this reason, it is often better to use a model with an upper bound instead of an exponential growth model, though the exponential growth model is still useful over a short term, before approaching the limiting value.
The logistic growth model is approximately exponential at first, but it has a reduced rate of growth as the output approaches the model’s upper bound, called the carrying capacity. For constants
 and
 the logistic growth of a population over time
 is represented by the model
The graph in  shows how the growth rate changes over time. The graph increases from left to right, but the growth rate only increases until it reaches its point of maximum growth rate, at which point the rate of increase decreases.

### Choosing an Appropriate Model for Data

Now that we have discussed various mathematical models, we need to learn how to choose the appropriate model for the raw data we have. Many factors influence the choice of a mathematical model, among which are experience, scientific laws, and patterns in the data itself. Not all data can be described by elementary functions. Sometimes, a function is chosen that approximates the data over a given interval. For instance, suppose data were gathered on the number of homes bought in the United States from the years 1960 to 2013. After plotting these data in a scatter plot, we notice that the shape of the data from the years 2000 to 2013 follow a logarithmic curve. We could restrict the interval from 2000 to 2010, apply regression analysis using a logarithmic model, and use it to predict the number of home buyers for the year 2015.
Three kinds of functions that are often useful in mathematical models are linear functions, exponential functions, and logarithmic functions. If the data lies on a straight line, or seems to lie approximately along a straight line, a linear model may be best. If the data is non-linear, we often consider an exponential or logarithmic model, though other models, such as quadratic models, may also be considered.
In choosing between an exponential model and a logarithmic model, we look at the way the data curves. This is called the concavity. If we draw a line between two data points, and all (or most) of the data between those two points lies above that line, we say the curve is concave down. We can think of it as a bowl that bends downward and therefore cannot hold water. If all (or most) of the data between those two points lies below the line, we say the curve is concave up. In this case, we can think of a bowl that bends upward and can therefore hold water. An exponential curve, whether rising or falling, whether representing growth or decay, is always concave up away from its horizontal asymptote. A logarithmic curve is always concave away from its vertical asymptote. In the case of positive data, which is the most common case, an exponential curve is always concave up, and a logarithmic curve always concave down.
A logistic curve changes concavity. It starts out concave up and then changes to concave down beyond a certain point, called a point of inflection.
After using the graph to help us choose a type of function to use as a model, we substitute points, and solve to find the parameters. We reduce round-off error by choosing points as far apart as possible.

### Expressing an Exponential Model in Base

While powers and logarithms of any base can be used in modeling, the two most common bases are
 and
 In science and mathematics, the base
 is often preferred. We can use laws of exponents and laws of logarithms to change any base to base

### Key Equations

### Key Concepts

1. The basic exponential function is
 If
 we have exponential growth; if
 we have exponential decay.
2. We can also write this formula in terms of continuous growth as
 where
 is the starting value. If
 is positive, then we have exponential growth when
 and exponential decay when
 See .
3. In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay. See .
4. We can find the age,
 of an organic artifact by measuring the amount,
 of carbon-14 remaining in the artifact and using the formula
 to solve for
 See .
5. Given a substance’s doubling time or half-time, we can find a function that represents its exponential growth or decay. See .
6. We can use Newton’s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time. See .
7. We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors. See .
8. We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data. See .
9. Any exponential function with the form
 can be rewritten as an equivalent exponential function with the form
 where
 See .

### Section Exercises

### Verbal

### Numeric

For the following exercises, use the logistic growth model

### Technology

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.
For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in
 years is modeled by the equation

### Extensions

### Real-World Applications

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.
For the following exercises, use this scenario: A tumor is injected with
 grams of Iodine-125, which has a decay rate of

 per day.
For the following exercises, use this scenario: A biologist recorded a count of
 bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes.
For the following exercises, use this scenario: A pot of warm soup with an internal temperature of
 Fahrenheit was taken off the stove to cool in a
 room. After fifteen minutes, the internal temperature of the soup was

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of
 and is allowed to cool in a
 room. After half an hour, the internal temperature of the turkey is

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.
For the following exercises, use this scenario: The equation
 models the number of people in a town who have heard a rumor after t days.
For the following exercise, choose the correct answer choice.


# Exponential and Logarithmic Functions

## Fitting Exponential Models to Data

### Learning Objectives

1. Draw and interpret scatter diagrams (linear, exponential, logarithmic). (CA 4.3.1)
2. Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions. (CA 4.3.4)

### Objective 1: Draw and interpret scatter diagrams (linear, exponential, logarithmic). (CA 4.3.1)

A Scatter Plot is a graph of plotted points that may show a relationship between the variables in a set of data.

### Practice Makes Perfect

Draw and interpret scatter diagrams ( linear, exponential, logarithmic).

### Objective 2: Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions. (CA 4.3.4)

We can find a linear function that fits the data in the previous problem by “eyeballing” a line that seems to fit. But while estimating a line works relatively well, technology can help us find a line that fits the data as perfect as possible.

This line is called the Least Squares Regression Line or Linear Regression Model.
A regression line is a line that is closest to the data in the scatter plot, which means that such a line is a best fit for the data.
Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions.

### Practice Makes Perfect

Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions.
In previous sections of this chapter, we were either given a function explicitly to graph or evaluate, or we were given a set of points that were guaranteed to lie on the curve. Then we used algebra to find the equation that fit the points exactly. In this section, we use a modeling technique called regression analysis to find a curve that models data collected from real-world observations. With regression analysis, we don’t expect all the points to lie perfectly on the curve. The idea is to find a model that best fits the data. Then we use the model to make predictions about future events.
Do not be confused by the word model. In mathematics, we often use the terms function, equation, and model interchangeably, even though they each have their own formal definition. The term model is typically used to indicate that the equation or function approximates a real-world situation.
We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review. Take a moment to rethink each of these functions, reflect on the work we’ve done so far, and then explore the ways regression is used to model real-world phenomena.

### Building an Exponential Model from Data

As we’ve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But that’s not the whole story. It’s the way data increase or decrease that helps us determine whether it is best modeled by an exponential equation. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let’s review exponential growth and decay.
Recall that exponential functions have the form
  or
  When performing regression analysis, we use the form most commonly used on graphing utilities,
  Take a moment to reflect on the characteristics we’ve already learned about the exponential function
  (assume

1.
  must be greater than zero and not equal to one.
2. The initial value of the model is

As part of the results, your calculator will display a number known as the correlation coefficient, labeled by the variable
  or
  (You may have to change the calculator’s settings for these to be shown.) The values are an indication of the “goodness of fit” of the regression equation to the data. We more commonly use the value of
  instead of
  but the closer either value is to 1, the better the regression equation approximates the data.

### Building a Logarithmic Model from Data

Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best.
Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics we’ve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic regression analysis, we use the form of the logarithmic function most commonly used on graphing utilities,
  For this function
1. All input values,
  must be greater than zero.
2. The point
  is on the graph of the model.
3. If
  the model is increasing. Growth increases rapidly at first and then steadily slows over time.
4. If
  the model is decreasing. Decay occurs rapidly at first and then steadily slows over time.

### Building a Logistic Model from Data

Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or limiting value. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients.
It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic regression analysis, we use the form most commonly used on graphing utilities:
Recall that:
1.
  is the initial value of the model.
2. when
  the model increases rapidly at first until it reaches its point of maximum growth rate,
  At that point, growth steadily slows and the function becomes asymptotic to the upper bound

3.

is the limiting value, sometimes called the carrying capacity, of the model.

### Key Concepts

1. Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.
2. We use the command “ExpReg” on a graphing utility to fit function of the form
  to a set of data points. See .
3. Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time.
4. We use the command “LnReg” on a graphing utility to fit a function of the form
  to a set of data points. See .
5. Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit.
6. We use the command “Logistic” on a graphing utility to fit a function of the form
  to a set of data points. See .

### Section Exercises

### Verbal

### Graphical

For the following exercises, match the given function of best fit with the appropriate scatterplot in  through . Answer using the letter beneath the matching graph.

### Numeric

### Technology

For the following exercises, use this scenario: The population
  of a koi pond over
  months is modeled by the function

For the following exercises, use this scenario: The population
  of an endangered species habitat for wolves is modeled by the function
  where
  is given in years.
For the following exercises, refer to .

For the following exercises, refer to .

For the following exercises, refer to .

For the following exercises, refer to .

For the following exercises, refer to .

For the following exercises, refer to .

### Extensions

### Chapter Review Exercises

### Exponential Functions

### Graphs of Exponential Functions

### Logarithmic Functions

### Graphs of Logarithmic Functions

### Logarithmic Properties

### Exponential and Logarithmic Equations

### Exponential and Logarithmic Models

For the following exercises, use this scenario: A doctor prescribes
  milligrams of a therapeutic drug that decays by about
  each hour.
For the following exercises, use this scenario: A soup with an internal temperature of
  Fahrenheit was taken off the stove to cool in a
  room. After fifteen minutes, the internal temperature of the soup was

For the following exercises, use this scenario: The equation
  models the number of people in a school who have heard a rumor after
  days.
For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

### Fitting Exponential Models to Data

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

### Practice Test

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.


# The Unit Circle: Sine and Cosine Functions

## Introduction to The Unit Circle: Sine and Cosine Functions

Life is dense with phenomena that repeat in regular intervals. Each day, for example, the tides rise and fall in response to the gravitational pull of the moon. And as a result of the motion of the moon itself, the tides occur with different strengths. Throughout history, many Indigenous peoples have used this regularity to build cultural narratives and direct key activities, such as agriculture, hunting, and fishing. Aboriginal people in the Torres Strait area (the northern tip) of Australia used the tidal peaks to determine the best times to fish. Their elders explain that the stronger spring tides stirred up sediment and obscured fish vision, leaving them more likely to take in lures and resulting in a larger catch.Hamacher, D.W., Tapim, A., Passi, S., and Barsa, J. (2018). Dancing with the stars – astronomy and music in the Torres Strait. In Imagining Other Worlds: Explorations in Astronomy and Culture.
In mathematics, a function that repeats its values in regular intervals is known as a periodic function. The graphs of such functions show a general shape reflective of a pattern that keeps repeating. This means the graph of the function has the same output at exactly the same place in every cycle. And this translates to all the cycles of the function having exactly the same length. So, if we know all the details of one full cycle of a true periodic function, then we know the state of the function’s outputs at all times, future and past. In this chapter, we will investigate various examples of periodic functions.


# The Unit Circle: Sine and Cosine Functions

## Angles

A golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure in many undertakings. In this section, we will examine properties of angles.

### Drawing Angles in Standard Position

Properly defining an angle first requires that we define a ray. A ray is a directed line segment. It consists of one point on a line and all points extending in one direction from that point. The first point is called the endpoint of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray in  can be named as ray EF, or in symbol form

An angle is the union of two rays having a common endpoint. The endpoint is called the vertex of the angle, and the two rays are the sides of the angle. The angle in  is formed from
  and
 . Angles can be named using a point on each ray and the vertex, such as angle DEF, or in symbol form

Greek letters are often used as variables for the measure of an angle.  is a list of Greek letters commonly used to represent angles, and a sample angle is shown in .
Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. In order to identify the different sides, we indicate the rotation with a small arrow close to the vertex as in .
As we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The measure of an angle is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One degree is
  of a circular rotation, so a complete circular rotation contains
  degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol
  For example,

To formalize our work, we will begin by drawing angles on an x-y coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis. See .
If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle. If the angle is measured in a clockwise direction, the angle is said to be a negative angle.
Drawing an angle in standard position always starts the same way—draw the initial side along the positive x-axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by
  For example, to draw a
  angle, we calculate that
  So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive x-axis. To draw a
  angle, we calculate that
  So the terminal side will be 1 complete rotation around the circle, moving counterclockwise from the positive x-axis. In this case, the initial side and the terminal side overlap. See .
Since we define an angle in standard position by its terminal side, we have a special type of angle whose terminal side lies on an axis, a quadrantal angle. This type of angle can have a measure of 0°, 90°, 180°, 270°, or  360°. See .

### Converting Between Degrees and Radians

Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle.
The circumference of a circle is
  If we divide both sides of this equation by
  we create the ratio of the circumference, which is always
  to the radius, regardless of the length of the radius. So the circumference of any circle is
  times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in .
This brings us to our new angle measure. One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals
  times the radius, a full circular rotation is
  radians.
See . Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel.

### Relating Arc Lengths to Radius

An arc length
  is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.
This ratio, called the radian measure, is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length
  to the radius r. See .
If
  then

To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is
  where
  is the radius. The smaller circle then has circumference
  and the larger has circumference
  Now we draw a
  angle on the two circles, as in .
Notice what happens if we find the ratio of the arc length divided by the radius of the circle.
Since both ratios are
  the angle measures of both circles are the same, even though the arc length and radius differ.

### Using Radians

Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in , suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure.
Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees,
  We can also track one rotation around a circle by finding the circumference,
  and for the unit circle
  These two different ways to rotate around a circle give us a way to convert from degrees to radians.

### Identifying Special Angles Measured in Radians

In addition to knowing the measurements in degrees and radians of a quarter revolution, a half  revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in . Memorizing these angles will be very useful as we study the properties associated with angles.
Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in , which are shown in . Be sure you can verify each of these measures.

### Converting Between Radians and Degrees

Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion where
  is the measure of the angle in degrees and
  is the measure of the angle in radians.
This proportion shows that the measure of angle
  in degrees divided by 180 equals the measure of angle
  in radians divided by
  Or, phrased another way, degrees is to 180 as radians is to

### Finding Coterminal Angles

Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of
  to
  or
  to
  It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution.
It is possible for more than one angle to have the same terminal side. Look at . The angle of
  is a positive angle, measured counterclockwise. The angle of
  is a negative angle, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles. Every angle greater than
  or less than
  is coterminal with an angle between
  and
  and it is often more convenient to find the coterminal angle within the range of
  to
  than to work with an angle that is outside that range.
Any angle has infinitely many coterminal angles because each time we add
   to that angle—or subtract
   from it—the resulting value has a terminal side in the same location. For example,
   and
   are coterminal for this reason, as is

An angle’s reference angle is the measure of the smallest, positive, acute angle
   formed by the terminal side of the angle
   and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See  for examples of reference angles for angles in different quadrants.

### Finding Coterminal Angles Measured in Radians

We can find coterminal angles measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.

### Determining the Length of an Arc

Recall that the radian measure
   of an angle was defined as the ratio of the arc length
   of a circular arc to the radius
   of the circle,
   From this relationship, we can find arc length along a circle, given an angle.

### Finding the Area of a Sector of a Circle

In addition to arc length, we can also use angles to find the area of a sector of a circle. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie. Recall that the area of a circle with radius
   can be found using the formula
   If the two radii form an angle of
   measured in radians, then
   is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the area of a sector is the fraction
   multiplied by the entire area. (Always remember that this formula only applies if
   is in radians.)

### Use Linear and Angular Speed to Describe Motion on a Circular Path

In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed is speed along a straight path and can be determined by the distance it moves along (its displacement) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or
   inches, every second. So the linear speed of the point is
   in./s. The equation for linear speed is as follows where
   is linear speed,
   is displacement, and
   is time.
Angular speed results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as
   90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where
   (read as omega) is angular speed,
   is the angle traversed, and
   is time.
Combining the definition of angular speed with the arc length equation,
   we can find a relationship between angular and linear speeds. The angular speed equation can be solved for
   giving
   Substituting this into the arc length equation gives:
Substituting this into the linear speed equation gives:
Water wheels have been used for thousands of years to transfer the power of flowing water to other devices. The image below depicts the design of the the 3rd century Roman water wheel in Hierapolis, a city in what is now Turkey. Water turned the wheel, which in turn rotated a crank connected to two saws used to cut blocks. These design elements were used in water wheel applications throughout the world, and even provided the underlying principle for the steam engine, invented about 1500 years later.

### Key Equations

### Key Concepts

1. An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. The amount of rotation determines the measure of the angle.
2. An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis. A positive angle is measured counterclockwise from the initial side and a negative angle is measured clockwise.
3. To draw an angle in standard position, draw the initial side along the positive x-axis and then place the terminal side according to the fraction of a full rotation the angle represents. See .
4. In addition to degrees, the measure of an angle can be described in radians. See .
5. To convert between degrees and radians, use the proportion
  See  and .
6. Two angles that have the same terminal side are called coterminal angles.
7. We can find coterminal angles by adding or subtracting
  or
  See  and .
8. Coterminal angles can be found using radians just as they are for degrees. See .
9. The length of a circular arc is a fraction of the circumference of the entire circle. See .
10. The area of sector is a fraction of the area of the entire circle. See .
11. An object moving in a circular path has both linear and angular speed.
12. The angular speed of an object traveling in a circular path is the measure of the angle through which it turns in a unit of time. See .
13. The linear speed of an object traveling along a circular path is the distance it travels in a unit of time. See .

### Section Exercises

### Verbal

### Graphical

For the following exercises, draw an angle in standard position with the given measure.
For the following exercises, refer to . Round to two decimal places.
For the following exercises, refer to . Round to two decimal places.

### Algebraic

For the following exercises, convert angles in radians to degrees.
For the following exercises, convert angles in degrees to radians.
For the following exercises, use the given information to find the length of a circular arc. Round to two decimal places.
For the following exercises, use the given information to find the area of the sector. Round to four decimal places.
For the following exercises, find the angle between
   and
   that is coterminal to the given angle.
For the following exercises, find the angle between 0 and
   in radians that is coterminal to the given angle.

### Real-World Applications

### Extensions


# The Unit Circle: Sine and Cosine Functions

## Right Triangle Trigonometry

Mt. Everest, which straddles the border between China and Nepal, is the tallest mountain in the world. Measuring its height is no easy task. In fact, the actual measurement has been a source of controversy for hundreds of years. The measurement process involves the use of triangles and a branch of mathematics known as trigonometry. In this section, we will define a new group of functions known as trigonometric functions, and find out how they can be used to measure heights, such as those of the tallest mountains.

### Using Right Triangles to Evaluate Trigonometric Functions

 shows a right triangle with a vertical side of length
 and a horizontal side has length
 Notice that the triangle is inscribed in a circle of radius 1. Such a circle, with a center at the origin and a radius of 1, is known as a unit circle.
We can define the trigonometric functions in terms an angle t and the lengths of the sides of the triangle. The adjacent side is the side closest to the angle, x. (Adjacent means “next to.”) The opposite side is the side across from the angle, y. The hypotenuse is the side of the triangle opposite the right angle, 1. These sides are labeled in .
Given a right triangle with an acute angle of
 the first three trigonometric functions are listed.
A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “ine is pposite over ypotenuse, osine is djacent over ypotenuse, angent is pposite over djacent.”
For the triangle shown in , we have the following.

### Reciprocal Functions

In addition to sine, cosine, and tangent, there are three more functions. These too are defined in terms of the sides of the triangle.
Take another look at these definitions. These functions are the reciprocals of the first three functions.
When working with right triangles, keep in mind that the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in . The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.
Many problems ask for all six trigonometric functions for a given angle in a triangle. A possible strategy to use is to find the sine, cosine, and tangent of the angles first. Then, find the other trigonometric functions easily using the reciprocals.

### Finding Trigonometric Functions of Special Angles Using Side Lengths

It is helpful to evaluate the trigonometric functions as they relate to the special angles—multiples of
 and
 Remember, however, that when dealing with right triangles, we are limited to angles between

Suppose we have a
 triangle, which can also be described as a
 triangle. The sides have lengths in the relation
 The sides of a
 triangle, which can also be described as a
 triangle, have lengths in the relation
 These relations are shown in .
We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.

### Using Equal Cofunction of Complements

If we look more closely at the relationship between the sine and cosine of the special angles, we notice a pattern. In a right triangle with angles of
 and
 we see that the sine of
 namely
 is also the cosine of
  while the sine of
 namely
 is also the cosine of

See .
This result should not be surprising because, as we see from , the side opposite the angle of
 is also the side adjacent to
 so
 and
 are exactly the same ratio of the same two sides,
 and
 Similarly,
 and
 are also the same ratio using the same two sides,
 and

The interrelationship between the sines and cosines of
 and
 also holds for the two acute angles in any right triangle, since in every case, the ratio of the same two sides would constitute the sine of one angle and the cosine of the other. Since the three angles of a triangle add to
 and the right angle is
 the remaining two angles must also add up to
 That means that a right triangle can be formed with any two angles that add to
 —in other words, any two complementary angles. So we may state a cofunction identity: If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. This identity is illustrated in .
Using this identity, we can state without calculating, for instance, that the sine of
 equals the cosine of
 and that the sine of
 equals the cosine of
 We can also state that if, for a given angle
 then
 as well.

### Using Trigonometric Functions

In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.

### Using Right Triangle Trigonometry to Solve Applied Problems

Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height.
Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. See .

### Key Equations

### Key Concepts

1. We can define trigonometric functions as ratios of the side lengths of a right triangle. See .
2. The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. See .
3. We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which they occur. See .
4. Any two complementary angles could be the two acute angles of a right triangle.
5. If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa. See .
6. We can use trigonometric functions of an angle to find unknown side lengths.
7. Select the trigonometric function representing the ratio of the unknown side to the known side. See .
8. Right-triangle trigonometry facilitates the measurement of inaccessible heights and distances.
9. The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are known. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, use cofunctions of complementary angles.
For the following exercises, find the lengths of the missing sides if side
     is opposite angle
     side
     is opposite angle
     and side
     is the hypotenuse.

### Graphical

For the following exercises, use  to evaluate each trigonometric function of angle

For the following exercises, use  to evaluate each trigonometric function of angle

For the following exercises, solve for the unknown sides of the given triangle.

### Technology

For the following exercises, use a calculator to find the length of each side to four decimal places.

### Extensions

### Real-World Applications


# The Unit Circle: Sine and Cosine Functions

## Unit Circle

Looking for a thrill? Then consider a ride on the Ain Dubai, the world's tallest Ferris wheel. Located in Dubai, the most populous city and the financial and tourism hub of the United Arab Emirates, the wheel soars to 820 feet, about 1.5 tenths of a mile. Described as an observation wheel, riders enjoy spectacular views of the Burj Khalifa (the world's tallest building) and the Palm Jumeirah (a human-made archipelago home to over 10,000 people and 20 resorts) as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs.

### Finding Trigonometric Functions Using the Unit Circle

We have already defined the trigonometric functions in terms of right triangles. In this section, we will redefine them in terms of the unit circle. Recall that a unit circle is a circle centered at the origin with radius 1, as shown in . The angle (in radians) that
 intercepts forms an arc of length
 Using the formula
 and knowing that
 we see that for a unit circle,

The x- and y-axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.
For any angle
 we can label the intersection of the terminal side and the unit circle as by its coordinates,
 The coordinates
 and
 will be the outputs of the trigonometric functions
  and
  respectively. This means
  and

### Defining Sine and Cosine Functions from the Unit Circle

The sine function relates a real number
 to the y-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle
 equals the y-value of the endpoint on the unit circle of an arc of length
 In , the sine is equal to
 Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the y-coordinate of the corresponding point on the unit circle.
The cosine function of an angle
 equals the x-value of the endpoint on the unit circle of an arc of length
 In , the cosine is equal to

Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses:
  is the same as
 and
 is the same as
 Likewise,
 is a commonly used shorthand notation for
 Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer.

### Finding Sines and Cosines of Angles on an Axis

For quadrantral angles, the corresponding point on the unit circle falls on the x- or y-axis. In that case, we can easily calculate cosine and sine from the values of
 and

### The Pythagorean Identity

Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is
 Because
  and
  we can substitute for
 and
 to get
 This equation,
 is known as the Pythagorean Identity. See .
We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution.

### Finding Sines and Cosines of Special Angles

We have already learned some properties of the special angles, such as the conversion from radians to degrees, and we found their sines and cosines using right triangles. We can also calculate sines and cosines of the special angles using the Pythagorean Identity.

### Finding Sines and Cosines of   Angles

First, we will look at angles of
 or
 as shown in .  A
 triangle is an isosceles triangle, so the x- and y-coordinates of the corresponding point on the circle are the same. Because the x- and y-values are the same, the sine and cosine values will also be equal.
At
 which is 45 degrees, the radius of the unit circle bisects the first quadrantal angle. This means the radius lies along the line
 A unit circle has a radius equal to 1 so the right triangle formed below the line
 has sides
 and
 and radius = 1. See .
From the Pythagorean Theorem we get
We can then substitute

Next we combine like terms.
And solving for
 we get
In quadrant I,

At
 or 45 degrees,
If we then rationalize the denominators, we get
Therefore, the
 coordinates of a point on a circle of radius
 at an angle of
 are

### Finding Sines and Cosines of   and   Angles

Next, we will find the cosine and sine at an angle of
 or
 First, we will draw a triangle inside a circle with one side at an angle of
 and another at an angle of
 as shown in . If the resulting two right triangles are combined into one large triangle, notice that all three angles of this larger triangle will be
 as shown in .
Because all the angles are equal, the sides are also equal. The vertical line has length
 and since the sides are all equal, we can also conclude that
 or
 Since

And since
 in our unit circle,
Using the Pythagorean Identity, we can find the cosine value.
The
 coordinates for the point on a circle of radius
 at an angle of
 are
 At
 the radius of the unit circle, 1, serves as the hypotenuse of a 30-60-90 degree right triangle,
 as shown in . Angle
 has measure
 At point
 we draw an angle
 with measure of
 We know the angles in a triangle sum to
 so the measure of angle
 is also
 Now we have an equilateral triangle. Because each side of the equilateral triangle
 is the same length, and we know one side is the radius of the unit circle, all sides must be of length 1.
The measure of angle
 is 30°. Angle
 is double angle
 so its measure is 60°.
 is the perpendicular bisector of
 so it cuts
 in half.  This means that
 is
 the radius, or
 Notice that
 is the x-coordinate of point
 which is at the intersection of the 60° angle and the unit circle. This gives us a triangle
 with hypotenuse of 1 and side
 of length

From the Pythagorean Theorem, we get
Substituting
 we get
Solving for
 we get
Since
 has the terminal side in quadrant I where the y-coordinate is positive, we choose
 the positive value.
At
 (60°), the
 coordinates for the point on a circle of radius
 at an angle of
 are
 so we can find the sine and cosine.
We have now found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of the unit circle.  summarizes these values.
 shows the common angles in the first quadrant of the unit circle.

### Using a Calculator to Find Sine and Cosine

To find the cosine and sine of angles other than the special angles, we turn to a computer or calculator. Be aware: Most calculators can be set into “degree” or “radian” mode, which tells the calculator the units for the input value. When we evaluate
 on our calculator, it will evaluate it as the cosine of 30 degrees if the calculator is in degree mode, or the cosine of 30 radians if the calculator is in radian mode.

### Identifying the Domain and Range of Sine and Cosine Functions

Now that we can find the sine and cosine of an angle, we need to discuss their domains and ranges. What are the domains of the sine and cosine functions? That is, what are the smallest and largest numbers that can be inputs of the functions? Because angles smaller than
 and angles larger than
 can still be graphed on the unit circle and have real values of
  and  there is no lower or upper limit to the angles that can be inputs to the sine and cosine functions. The input to the sine and cosine functions is the rotation from the positive x-axis, and that may be any real number.
What are the ranges of the sine and cosine functions? What are the least and greatest possible values for their output? We can see the answers by examining the unit circle, as shown in . The bounds of the x-coordinate are
 The bounds of the y-coordinate are also
 Therefore, the range of both the sine and cosine functions is

### Finding Reference Angles

We have discussed finding the sine and cosine for angles in the first quadrant, but what if our angle is in another quadrant? For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value. Because the sine value is the y-coordinate on the unit circle, the other angle with the same sine will share the same y-value, but have the opposite x-value. Therefore, its cosine value will be the opposite of the first angle’s cosine value.
Likewise, there will be an angle in the fourth quadrant with the same cosine as the original angle. The angle with the same cosine will share the same x-value but will have the opposite y-value. Therefore, its sine value will be the opposite of the original angle’s sine value.
As shown in , angle
 has the same sine value as angle
 the cosine values are opposites. Angle
 has the same cosine value as angle
 the sine values are opposites.
Recall that an angle’s reference angle is the acute angle,
 formed by the terminal side of the angle
 and the horizontal axis. A reference angle is always an angle between
 and
 or
 and
 radians. As we can see from , for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.

### Using Reference Angles

Now let’s take a moment to reconsider the Ferris wheel introduced at the beginning of this section. Suppose a rider snaps a photograph while stopped twenty feet above ground level. The rider then rotates three-quarters of the way around the circle. What is the rider’s new elevation? To answer questions such as this one, we need to evaluate the sine or cosine functions at angles that are greater than 90 degrees or at a negative angle. Reference angles make it possible to evaluate trigonometric functions for angles outside the first quadrant. They can also be used to find
 coordinates for those angles. We will use the reference angle of the angle of rotation combined with the quadrant in which the terminal side of the angle lies.

### Using Reference Angles to Evaluate Trigonometric Functions

We can find the cosine and sine of any angle in any quadrant if we know the cosine or sine of its reference angle. The absolute values of the cosine and sine of an angle are the same as those of the reference angle. The sign depends on the quadrant of the original angle. The cosine will be positive or negative depending on the sign of the x-values in that quadrant. The sine will be positive or negative depending on the sign of the y-values in that quadrant.

### Using Reference Angles to Find Coordinates

Now that we have learned how to find the cosine and sine values for special angles in the first quadrant, we can use symmetry and reference angles to fill in cosine and sine values for the rest of the special angles on the unit circle. They are shown in . Take time to learn the
  coordinates of all of the major angles in the first quadrant.
In addition to learning the values for special angles, we can use reference angles to find
 coordinates of any point on the unit circle, using what we know of reference angles along with the identities
First we find the reference angle corresponding to the given angle. Then we take the sine and cosine values of the reference angle, and give them the signs corresponding to the y- and x-values of the quadrant.

### Key Equations

### Key Concepts

1. Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit.
2. Using the unit circle, the sine of an angle
 equals the y-value of the endpoint on the unit circle of an arc of length
 whereas the cosine of an angle
 equals the x-value of the endpoint. See .
3. The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. See .
4. When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles. See .
5. Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. See .
6. The domain of the sine and cosine functions is all real numbers.
7. The range of both the sine and cosine functions is

8. The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.
9. The signs of the sine and cosine are determined from the x- and y-values in the quadrant of the original angle.
10. An angle’s reference angle is the size angle,
 formed by the terminal side of the angle
 and the horizontal axis. See .
11. Reference angles can be used to find the sine and cosine of the original angle. See .
12. Reference angles can also be used to find the coordinates of a point on a circle. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by
         lies.
For the following exercises, find the exact value of each trigonometric function.

### Numeric

For the following exercises, state the reference angle for the given angle.
For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.
For the following exercises, find the requested value.

### Graphical

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of

### Technology

For the following exercises, use a graphing calculator to evaluate.

### Extensions

For the following exercises, evaluate.

### Real-World Applications

For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point
  that is, on the due north position.  Assume the carousel revolves counter clockwise.


# The Unit Circle: Sine and Cosine Functions

## The Other Trigonometric Functions

A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is
 or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. We have already defined the sine and cosine functions of an angle. Though sine and cosine are the trigonometric functions most often used, there are four others. Together they make up the set of six trigonometric functions. In this section, we will investigate the remaining functions.

### Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent

We can also define the remaining functions in terms of the unit circle with a point
 corresponding to an angle of
  as shown in . As with the sine and cosine, we can use the
 coordinates to find the other functions.
The first function we will define is the tangent. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle. In , the tangent of angle
 is equal to
 Because the y-value is equal to the sine of
  and the x-value is equal to the cosine of
 the tangent of angle
 can also be defined as
  The tangent function is abbreviated as
 The remaining three functions can all be expressed as reciprocals of functions we have already defined.
1. The secant function is the reciprocal of the cosine function. In , the secant of angle
 is equal to
  The secant function is abbreviated as

2. The cotangent function is the reciprocal of the tangent function. In , the cotangent of angle
 is equal to
  The cotangent function is abbreviated as

3. The cosecant function is the reciprocal of the sine function. In , the cosecant of angle
 is equal to
  The cosecant function is abbreviated as

Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting
 equal to the cosine and
 equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in .

### Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent

We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x- and y-values in the original quadrant.  shows which functions are positive in which quadrant.
To help remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “A,” ll of the six trigonometric functions are positive. In quadrant II, “Smart,” only ine and its reciprocal function, cosecant, are positive.   In quadrant III, “Trig,” only angent and its reciprocal function, cotangent, are positive.  Finally, in quadrant IV, “Class,” only osine and its reciprocal function, secant, are positive.

### Using Even and Odd Trigonometric Functions

To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.
Consider the function
 shown in . The graph of the function is symmetrical about the y-axis. All along the curve, any two points with opposite x-values have the same function value. This matches the result of calculation:
 and so on. So
 is an even function, a function such that two inputs that are opposites have the same output. That means

Now consider the function
 shown in . The graph is not symmetrical about the y-axis. All along the graph, any two points with opposite x-values also have opposite y-values. So
 is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means

We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in . The sine of the positive angle is
 The sine of the negative angle is
 The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in .

### Recognizing and Using Fundamental Identities

We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.

### Alternate Forms of the Pythagorean Identity

We can use these fundamental identities to derive alternate forms of the Pythagorean Identity,
         One form is obtained by dividing both sides by

The other form is obtained by dividing both sides by

As we discussed at the beginning of the chapter, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or
         will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.
Other functions can also be periodic. For example, the lengths of months repeat every four years. If
         represents the length time, measured in years, and
         represents the number of days in February, then
         This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.

### Evaluating Trigonometric Functions with a Calculator

We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.
Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.
If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor
  to convert the degrees to radians. To find the secant of
  we could press

### Key Equations

### Key Concepts

1. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle.
2. The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
3. The six trigonometric functions can be found from a point on the unit circle. See .
4. Trigonometric functions can also be found from an angle. See  .
5. Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See .
6. A function is said to be even if
  and odd if
  for all x in the domain of f.
7. Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
8. Even and odd properties can be used to evaluate trigonometric functions. See .
9. The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
10. Identities can be used to evaluate trigonometric functions. See  and .
11. Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See .
12. The trigonometric functions repeat at regular intervals.
13. The period
  of a repeating function
  is the smallest interval such that
  for any value of

14. The values of trigonometric functions can be found by mathematical analysis. See  and .
15. To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the exact value of each expression.
For the following exercises, use reference angles to evaluate the expression.

### Graphical

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

### Technology

For the following exercises, use a graphing calculator to evaluate to three decimal places.

### Extensions

For the following exercises, use identities to evaluate the expression.
For the following exercises, use identities to simplify the expression.

### Real-World Applications

### Chapter Review Exercises

### Angles

For the following exercises, convert the angle measures to degrees.
For the following exercises, convert the angle measures to radians.
For the following exercises, find the angle between
  and
  that is coterminal with the given angle.
For the following exercises, find the angle between 0 and
  in radians that is coterminal with the given angle.
For the following exercises, draw the angle provided in standard position on the Cartesian plane.

### Right Triangle Trigonometry

For the following exercises, use side lengths to evaluate.
For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.
For the following exercises, use  to evaluate each trigonometric function.
For the following exercises, solve for the unknown sides of the given triangle.

### Unit Circle

### The Other Trigonometric Functions

For the following exercises, find the exact value of the given expression.
For the following exercises, use reference angles to evaluate the given expression.

### Chapter Practice Test


# Periodic Functions

## Introduction to Periodic Functions

The sun has played a core role in many religions. The ancient Egyptian culture portrayed the sun god, Ra (sometimes written as Re), as undertaking a two-part daily journey, with one portion in the sky (day) and the other through the underworld (night). Surya, the Hindu sun god, traces a similar path through the sky on a chariot pulled by seven horses. While their origins and associated narratives are quite different, both Ra and Surya are primary deities and seen as creators and preservers of life. In many Native American cultures, the sun is core to spiritual and religious practice, but is not always a deity. The Sun Dance, practiced differently by many Native American tribes, was a ceremony that generally paid homage to the sun and, in many cases, tested or expressed the strength of the tribe's people.
As one of the most most prominent natural phenomena and with its close association to giving life, the sun was an obvious subject for reverence. And its regularity, even in ancient times, made it the primary determinant of time. Each day, the sun rises in an easterly direction, approaches some maximum height relative to the celestial equator, and sets in a westerly direction. The celestial equator is an imaginary line that divides the visible universe into two halves in much the same way Earth’s equator is an imaginary line that divides the planet into two halves. The exact path the sun appears to follow depends on the exact location on Earth, but each location observes a predictable pattern over time.
The pattern of the sun’s motion throughout the course of a year is a periodic function. Creating a visual representation of a periodic function in the form of a graph can help us analyze the properties of the function. In this chapter, we will investigate graphs of sine, cosine, and other trigonometric functions.


# Periodic Functions

## Graphs of the Sine and Cosine Functions

White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.
Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions.

### Graphing Sine and Cosine Functions

Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function. We can create a table of values and use them to sketch a graph.  lists some of the values for the sine function on a unit circle.
Plotting the points from the table and continuing along the x-axis gives the shape of the sine function. See .
Notice how the sine values are positive between 0 and
  which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between
  and
  which correspond to the values of the sine function in quadrants III and IV on the unit circle. See .
Now let’s take a similar look at the cosine function. Again, we can create a table of values and use them to sketch a graph.  lists some of the values for the cosine function on a unit circle.
As with the sine function, we can plots points to create a graph of the cosine function as in .
Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval

In both graphs, the shape of the graph repeats after
  which means the functions are periodic with a period of
  A periodic function is a function for which a specific horizontal shift, P, results in a function equal to the original function:
  for all values of
  in the domain of
  When this occurs, we call the smallest such horizontal shift with
  the period of the function.  shows several periods of the sine and cosine functions.
Looking again at the sine and cosine functions on a domain centered at the y-axis helps reveal symmetries. As we can see in , the sine function is symmetric about the origin. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because

Now we can clearly see this property from the graph.
 shows that the cosine function is symmetric about the y-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that

### Investigating Sinusoidal Functions

As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. The general forms of sinusoidal functions are

### Determining the Period of Sinusoidal Functions

Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.
In the general formula,
  is related to the period by
  If
  then the period is less than
  and the function undergoes a horizontal compression, whereas if
  then the period is greater than
  and the function undergoes a horizontal stretch. For example,

  so the period is
  which we knew. If
  then
  so the period is
  and the graph is compressed. If
  then
  so the period is
  and the graph is stretched. Notice in  how the period is indirectly related to

### Determining Amplitude

Returning to the general formula for a sinusoidal function, we have analyzed how the variable
  relates to the period. Now let’s turn to the variable
  so we can analyze how it is related to the amplitude, or greatest distance from rest.
  represents the vertical stretch factor, and its absolute value
  is the amplitude. The local maxima will be a distance
  above the horizontal midline of the graph, which is the line
  because
  in this case, the midline is the x-axis. The local minima will be the same distance below the midline. If
  the function is stretched. For example, the amplitude of
  is twice the amplitude of
  If
  the function is compressed.  compares several sine functions with different amplitudes.

### Analyzing Graphs of Variations of y = sin x and y = cos x

Now that we understand how
  and
  relate to the general form equation for the sine and cosine functions, we will explore the variables
  and
  Recall the general form:
The value
  for a sinusoidal function is called the phase shift, or the horizontal displacement of the basic sine or cosine function. If
  the graph shifts to the right. If
  the graph shifts to the left. The greater the value of
  the more the graph is shifted.  shows that the graph of
  shifts to the right by
  units, which is more than we see in the graph of
  which shifts to the right by
  units.
While
  relates to the horizontal shift,
  indicates the vertical shift from the midline in the general formula for a sinusoidal function. See . The function
  has its midline at

Any value of
  other than zero shifts the graph up or down.  compares
  with
  which is shifted 2 units up on a graph.

### Graphing Variations of y = sin x and y = cos x

Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.
Instead of focusing on the general form equations
we will let
  and
  and work with a simplified form of the equations in the following examples.

### Using Transformations of Sine and Cosine Functions

We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function.

### Key Equations

### Key Concepts

1. Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of

2. The function
  is odd, so its graph is symmetric about the origin. The function
  is even, so its graph is symmetric about the y-axis.
3. The graph of a sinusoidal function has the same general shape as a sine or cosine function.
4. In the general formula for a sinusoidal function, the period is
  See .
5. In the general formula for a sinusoidal function,
  represents amplitude. If
  the function is stretched, whereas if
  the function is compressed. See .
6. The value
  in the general formula for a sinusoidal function indicates the phase shift. See .
7. The value
  in the general formula for a sinusoidal function indicates the vertical shift from the midline. See .
8. Combinations of variations of sinusoidal functions can be detected from an equation. See .
9. The equation for a sinusoidal function can be determined from a graph. See  and .
10. A function can be graphed by identifying its amplitude and period. See  and .
11. A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See .
12. Sinusoidal functions can be used to solve real-world problems. See , , and .

### Section Exercises

### Verbal

### Graphical

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for
  Round answers to two decimal places if necessary.
For the following exercises, graph one full period of each function, starting at
  For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for
  State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

### Algebraic

For the following exercises, let

For the following exercises, let

### Technology

### Real-World Applications