# Periodic Functions

## Graphs of the Other Trigonometric Functions

We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure repeated occurrences of distance? Imagine, for example, a fire truck parked next to a warehouse. The rotating light from the truck would travel across the wall of the warehouse in regular intervals. If the input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at regular intervals. The tangent function can be used to approximate this distance. Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this section, we will explore the graphs of the tangent and other trigonometric functions.

### Analyzing the Graph of y = tan x

We will begin with the graph of the tangent function, plotting points as we did for the sine and cosine functions. Recall that
The period of the tangent function is
  because the graph repeats itself on intervals of
  where
  is a constant. If we graph the tangent function on
  to
  we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.
We can determine whether tangent is an odd or even function by using the definition of tangent.
Therefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in .
These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when
  we can use a table to look for a trend. Because
  and
  we will evaluate
  at radian measures
  as shown in .
As
  approaches
  the outputs of the function get larger and larger. Because
  is an odd function, we see the corresponding table of negative values in .
We can see that, as
  approaches
  the outputs get smaller and smaller. Remember that there are some values of
  for which
  For example,
  and
  At these values, the tangent function is undefined, so the graph of
  has discontinuities at
  At these values, the graph of the tangent has vertical asymptotes.  represents the graph of
  The tangent is positive from 0 to
  and from
  to
  corresponding to quadrants I and III of the unit circle.

### Graphing Variations of y = tan x

As with the sine and cosine functions, the tangent function can be described by a general equation.
We can identify horizontal and vertical stretches and compressions using values of
  and
  The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.
Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase stretching/compressing factor when referring to the constant

### Graphing One Period of a Stretched or Compressed Tangent Function

We can use what we know about the properties of the tangent function to quickly sketch a graph of any stretched and/or compressed tangent function of the form
  We focus on a single period of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function’s domain if we wish. Our limited domain is then the interval
  and the graph has vertical asymptotes at
  where
  On
  the graph will come up from the left asymptote at
  cross through the origin, and continue to increase as it approaches the right asymptote at
  To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use
because

### Graphing One Period of a Shifted Tangent Function

Now that we can graph a tangent function that is stretched or compressed, we will add a vertical and/or horizontal (or phase) shift. In this case, we add
  and
  to the general form of the tangent function.
The graph of a transformed tangent function is different from the basic tangent function
  in several ways:

### Analyzing the Graphs of y = sec x and y = cscx

The secant was defined by the reciprocal identity
  Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at

  etc. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.
We can graph
  by observing the graph of the cosine function because these two functions are reciprocals of one another. See . The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the secant function increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.
The secant graph has vertical asymptotes at each value of
  where the cosine graph crosses the x-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.
Note that, because cosine is an even function, secant is also an even function. That is,

As we did for the tangent function, we will again refer to the constant
  as the stretching factor, not the amplitude.
Similar to the secant, the cosecant is defined by the reciprocal identity
  Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at

  etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.
We can graph
  by observing the graph of the sine function because these two functions are reciprocals of one another. See . The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the cosecant function increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.
The cosecant graph has vertical asymptotes at each value of
  where the sine graph crosses the x-axis; we show these in the graph below with dashed vertical lines.
Note that, since sine is an odd function, the cosecant function is also an odd function. That is,

The graph of cosecant, which is shown in , is similar to the graph of secant.

### Graphing Variations of y = sec x and y= csc x

For shifted, compressed, and/or stretched versions of the secant and cosecant functions, we can follow similar methods to those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the cosecant function in the same way as for the secant and other functions.The equations become the following.

### Analyzing the Graph of y = cot x

The last trigonometric function we need to explore is cotangent. The cotangent is defined by the reciprocal identity
    Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at
    etc. Since the output of the tangent function is all real numbers, the output of the cotangent function is also all real numbers.
We can graph
    by observing the graph of the tangent function because these two functions are reciprocals of one another. See . Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.
The cotangent graph has vertical asymptotes at each value of
    where
    we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent,
    has vertical asymptotes at all values of
    where
    and
    at all values of
    where
    has its vertical asymptotes.

### Graphing Variations of y = cot x

We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.

### Using the Graphs of Trigonometric Functions to Solve Real-World Problems

Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let’s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a fire truck and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function.

### Key Equations

### Key Concepts

1. The tangent function has period

2.
  	   is a tangent with vertical and/or horizontal stretch/compression and shift. See , , and .
3. The secant and cosecant are both periodic functions with a period of

  	   gives a shifted, compressed, and/or stretched secant function graph. See  and .
4.
  	   gives a shifted, compressed, and/or stretched cosecant function graph. See  and .
5. The cotangent function has period
  	   and vertical asymptotes at

6. The range of cotangent is
  	   and the function is decreasing at each point in its range.
7. The cotangent is zero at

8.
  is a cotangent with vertical and/or horizontal stretch/compression and shift. See  and .
9. Real-world scenarios can be solved using graphs of trigonometric functions. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, match each trigonometric function with one of the following graphs.
For the following exercises, find the period and horizontal shift of each of the functions.
For the following exercises, evaluate the transformed functions.
For the following exercises, rewrite each expression such that the argument
     is positive.

### Graphical

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.
For the following exercises, find and graph two periods of the periodic function with the given stretching factor,
     period, and phase shift.
For the following exercises, find an equation for the graph of each function.

### Technology

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input
     as

### Real-World Applications


# Periodic Functions

## Inverse Trigonometric Functions

For any right triangle, given one other angle and the length of one side, we can figure out what the other angles and sides are. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of sides to an angle. This is where the notion of an inverse to a trigonometric function comes into play. In this section, we will explore the inverse trigonometric functions.

### Understanding and Using the Inverse Sine, Cosine, and Tangent Functions

In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in .
For example, if
       then we would write
       Be aware that
       does not mean
       The following examples illustrate the inverse trigonometric functions:
1. Since
  then

2. Since
         then

3. Since
         then

In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that, for a one-to-one function, if
  then an inverse function would satisfy

Bear in mind that the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. We choose a domain for each function that includes the number 0.  shows the graph of the sine function limited to
  and the graph of the cosine function limited to

 shows the graph of the tangent function limited to

These conventional choices for the restricted domain are somewhat arbitrary, but they have important, helpful characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible. The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next instead of being divided into two parts by an asymptote.
On these restricted domains, we can define the inverse trigonometric functions.
1. The inverse sine function
  means
  The inverse sine function is sometimes called the arcsine function, and notated

2. The inverse cosine function
  means
  The inverse cosine function is sometimes called the arccosine function, and notated

3. The inverse tangent function
  means
  The inverse tangent function is sometimes called the arctangent function, and notated

The graphs of the inverse functions are shown in , , and . Notice that the output of each of these inverse functions is a number, an angle in radian measure. We see that
  has domain
  and range

  has domain
  and range
  and
  has domain of all real numbers and range
  To find the domain and range of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line

### Finding the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent Functions

Now that we can identify inverse functions, we will learn to evaluate them. For most values in their domains, we must evaluate the inverse trigonometric functions by using a calculator, interpolating from a table, or using some other numerical technique. Just as we did with the original trigonometric functions, we can give exact values for the inverse functions when we are using the special angles, specifically
  (30°),
  (45°), and
  (60°), and their reflections into other quadrants.

### Using a Calculator to Evaluate Inverse Trigonometric Functions

To evaluate inverse trigonometric functions that do not involve the special angles discussed previously, we will need to use a calculator or other type of technology. Most scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse sine, cosine, and tangent functions. These may be labeled, for example, SIN

 , ARCSIN, or ASIN.
In the previous chapter, we worked with trigonometry on a right triangle to solve for the sides of a triangle given one side and an additional angle. Using the inverse trigonometric functions, we can solve for the angles of a right triangle given two sides, and we can use a calculator to find the values to several decimal places.
In these examples and exercises, the answers will be interpreted as angles and we will use
  as the independent variable. The value displayed on the calculator may be in degrees or radians, so be sure to set the mode appropriate to the application.

### Finding Exact Values of Composite Functions with Inverse Trigonometric Functions

There are times when we need to compose a trigonometric function with an inverse trigonometric function. In these cases, we can usually find exact values for the resulting expressions without resorting to a calculator. Even when the input to the composite function is a variable or an expression, we can often find an expression for the output. To help sort out different cases, let
    and
    be two different trigonometric functions belonging to the set
    and let
    and
    be their inverses.

### Evaluating Compositions of the Form f(f−1(y)) and f−1(f(x))

For any trigonometric function,
     for all
     in the proper domain for the given function. This follows from the definition of the inverse and from the fact that the range of
     was defined to be identical to the domain of
     However, we have to be a little more careful with expressions of the form

### Evaluating Compositions of the Form f−1(g(x))

Now that we can compose a trigonometric function with its inverse, we can explore how to evaluate a composition of a trigonometric function and the inverse of another trigonometric function. We will begin with compositions of the form
     For special values of
     we can exactly evaluate the inner function and then the outer, inverse function. However, we can find a more general approach by considering the relation between the two acute angles of a right triangle where one is
     making the other
     Consider the sine and cosine of each angle of the right triangle in .
Because
     we have
     if
     If
     is not in this domain, then we need to find another angle that has the same cosine as
     and does belong to the restricted domain; we then subtract this angle from
     Similarly,
     so
     if
     These are just the function-cofunction relationships presented in another way.

### Evaluating Compositions of the Form f(g−1(x))

To evaluate compositions of the form
     where
     and
     are any two of the functions sine, cosine, or tangent and
     is any input in the domain of
     we have exact formulas, such as
     When we need to use them, we can derive these formulas by using the trigonometric relations between the angles and sides of a right triangle, together with the use of Pythagoras’s relation between the lengths of the sides. We can use the Pythagorean identity,
     to solve for one when given the other. We can also use the inverse trigonometric functions to find compositions involving algebraic expressions.

### Key Concepts

1. An inverse function is one that “undoes” another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
2. Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains.
3. For any trigonometric function
     if
     then
     However,
     only implies
     if
     is in the restricted domain of
     See .
4. Special angles are the outputs of inverse trigonometric functions for special input values; for example,
  See .
5. A calculator will return an angle within the restricted domain of the original trigonometric function. See .
6. Inverse functions allow us to find an angle when given two sides of a right triangle. See .
7. In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example,
     See .
8. If the inside function is a trigonometric function, then the only possible combinations are
     if
     and
     if
      See  and .
9. When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function. See .
10. When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, evaluate the expressions.
For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.
For the following exercises, find the angle
     in the given right triangle. Round answers to the nearest hundredth.
For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.
For the following exercises, find the exact value of the expression in terms of

    with the help of a reference triangle.

### Extensions

For the following exercises, evaluate the expression without using a calculator. Give the exact value.
For the following exercises, find the function if

### Graphical

### Real-World Applications

### Chapter Review Exercises

### Graphs of the Sine and Cosine Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

### Graphs of the Other Trigonometric Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period,
midline equation, and asymptotes.
For the following exercises, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.
For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function:
  where the domain is the years since 1980 and the range is the population of the city.
For the following exercises, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.

### Inverse Trigonometric Functions

For the following exercises, find the exact value without the aid of a calculator.

### Chapter Practice Test

For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.
For the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the function.
For the following exercises, find the amplitude, period, phase shift, and midline.
For the following exercises, find the period and horizontal shift of each function.
For the following exercises, graph the functions on the specified window and answer the questions.
For the following exercises, let

For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.
For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.
For the following exercises, find the exact value.
For the following exercises, suppose

Evaluate the following expressions.

For the following exercises, determine whether the equation is true or false.


# Trigonometric Identities and Equations

## Introduction to Trigonometric Identities and Equations

When we think of tennis as a game of angles, we may imagine players racing up to the net, creating options to deliver powerful cross shots that will leave their opponent stumbling toward the line. This is an exciting and effective method of play, though it brings greater risk.
But while the excitement of the game interplays with all types of geometry,  some of the newest innovations make even more use of mathematics. With balls traveling well over 100 miles per hour judges cannot always discern the centimeter or millimeters of difference between a ball that is in or out of bounds. Professional tennis was among the first sports to rely on an advanced tracking system called Hawk-Eye to help make close calls. The system uses several high-resolution cameras that are able to monitor and the ball's movement and its position on the court. Using the images from several cameras at once, the system's computers use trigonometric calculations to triangulate the ball's exact position and, essentially, turn a series of two-dimensional images into a three-dimensional one. Also, since the ball travels faster than the cameras' frame rate, the system also must make predictions to show where a ball is at all times. These technologies generally provide a more accurate game that builds more confidence and fairness. Similar technologies are used for baseball, and automated strike-calling is under discussion.


# Trigonometric Identities and Equations

## Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions

In espionage movies, we see international spies with multiple passports, each claiming a different identity. However, we know that each of those passports represents the same person. The trigonometric identities act in a similar manner to multiple passports—there are many ways to represent the same trigonometric expression. Just as a spy will choose an Italian passport when traveling to Italy, we choose the identity that applies to the given scenario when solving a trigonometric equation.
In this section, we will begin an examination of the fundamental trigonometric identities, including how we can verify them and how we can use them to simplify trigonometric expressions.

### Verifying the Fundamental Trigonometric Identities

Identities enable us to simplify complicated expressions. They are the basic tools of trigonometry used in solving trigonometric equations, just as factoring, finding common denominators, and using special formulas are the basic tools of solving algebraic equations. In fact, we use algebraic techniques constantly to simplify trigonometric expressions. Basic properties and formulas of algebra, such as the difference of squares formula and the perfect squares formula, will simplify the work involved with trigonometric expressions and equations. We already know that all of the trigonometric functions are related because they all are defined in terms of the unit circle. Consequently, any trigonometric identity can be written in many ways.
To verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially rewrite the expression until it has been transformed into the same expression as the other side of the equation. Sometimes we have to factor expressions, expand expressions, find common denominators, or use other algebraic strategies to obtain the desired result. In this first section, we will work with the fundamental identities: the Pythagorean identities, the even-odd identities, the reciprocal identities, and the quotient identities.
We will begin with the Pythagorean identities (see ), which are equations involving trigonometric functions based on the properties of a right triangle. We have already seen and used the first of these identifies, but now we will also use additional identities.
The second and third identities can be obtained by manipulating the first. The identity
 is found by rewriting the left side of the equation in terms of sine and cosine.
 Prove:

Similarly,
 can be obtained by rewriting the left side of this identity in terms of sine and cosine. This gives
Recall that we determined which trigonometric functions are odd and which are even. The next set of fundamental identities is the set of even-odd identities. The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle. (See ).
Recall that an odd function is one in which
 for all
 in the domain of
 The sine function is an odd function because
 The graph of an odd function is symmetric about the origin. For example, consider corresponding inputs of
 and
 The output of
 is opposite the output of
 Thus,
This is shown in .
Recall that an even function is one in which
The graph of an even function is symmetric about the y-axis. The cosine function is an even function because

For example, consider corresponding inputs
 and
 The output of
 is the same as the output of
 Thus,
See .
For all
 in the domain of the sine and cosine functions, respectively, we can state the following:
1. Since
 sine is an odd function.
2. Since,
 cosine is an even function.
The other even-odd identities follow from the even and odd nature of the sine and cosine functions. For example, consider the tangent identity,
 We can interpret the tangent of a negative angle as
 Tangent is therefore an odd function, which means that
 for all
 in the domain of the tangent function.
The cotangent identity,
 also follows from the sine and cosine identities. We can interpret the cotangent of a negative angle as
 Cotangent is therefore an odd function, which means that
 for all
 in the domain of the cotangent function.
The cosecant function is the reciprocal of the sine function, which means that the cosecant of a negative angle will be interpreted as
 The cosecant function is therefore odd.
Finally, the secant function is the reciprocal of the cosine function, and the secant of a negative angle is interpreted as
 The secant function is therefore even.
To sum up, only two of the trigonometric functions, cosine and secant, are even. The other four functions are odd, verifying the even-odd identities.
The next set of fundamental identities is the set of reciprocal identities, which, as their name implies, relate trigonometric functions that are reciprocals of each other. See . Recall that we first encountered these identities when defining trigonometric functions from right angles in Right Angle Trigonometry.
The final set of identities is the set of quotient identities, which define relationships among certain trigonometric functions and can be very helpful in verifying other identities. See .
The reciprocal and quotient identities are derived from the definitions of the basic trigonometric functions.

### Using Algebra to Simplify Trigonometric Expressions

We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.
For example, the equation
 resembles the equation
 which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric expressions or equations.
Another example is the difference of squares formula,
 which is widely used in many areas other than mathematics, such as engineering, architecture, and physics. We can also create our own identities by continually expanding an expression and making the appropriate substitutions. Using algebraic properties and formulas makes many trigonometric equations easier to understand and solve.

### Key Equations

### Key Concepts

1. There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem.
2. Graphing both sides of an identity will verify it. See .
3. Simplifying one side of the equation to equal the other side is another method for verifying an identity. See  and .
4. The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the more complex side of the equation. See .
5. We can create an identity and then verify it. See .
6. Verifying an identity may involve algebra with the fundamental identities. See  and .
7. Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout this text, as they consist of the fundamental rules of mathematics. See , , and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, use the fundamental identities to fully simplify the expression.
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.
For the following exercises, verify the identity.

### Extensions

For the following exercises, prove or disprove the identity.
For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.


# Trigonometric Identities and Equations

## Sum and Difference Identities

How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities, commonly used in mathematical proofs, have had real-world applications for centuries, including their use in calculating long distances.
The trigonometric identities we will examine in this section can be traced to a Persian astronomer who lived around 950 AD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. These are special equations or postulates, true for all values input to the equations, and with innumerable applications.
In this section, we will learn techniques that will enable us to solve problems such as the ones presented above. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the term formula is used synonymously with the word identity.

### Using the Sum and Difference Formulas for Cosine

Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the special angles, which we can review in the unit circle shown in .
We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. See .
First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. See . Point
       is at an angle
       from the positive x-axis with coordinates
       and point
       is at an angle of
       from the positive x-axis with coordinates
       Note the measure of angle
       is

Label two more points:
       at an angle of
       from the positive x-axis with coordinates
       and point
       with coordinates
       Triangle
       is a rotation of triangle
       and thus the distance from
       to
       is the same as the distance from
       to

We can find the distance from
       to
       using the distance formula.
Then we apply the Pythagorean identity and simplify.
Similarly, using the distance formula we can find the distance from
       to

Applying the Pythagorean identity and simplifying we get:
Because the two distances are the same, we set them equal to each other and simplify.
Finally we subtract
       from both sides and divide both sides by

Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.

### Using the Sum and Difference Formulas for Sine

The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.

### Using the Sum and Difference Formulas for Tangent

Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.
Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall,

Let’s derive the sum formula for tangent.
We can derive the difference formula for tangent in a similar way.

### Using Sum and Difference Formulas for Cofunctions

Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall from Right Triangle Trigonometry that, if the sum of two positive angles is
       those two angles are complements, and the sum of the two acute angles in a right triangle is
       so they are also complements. In , notice that if one of the acute angles is labeled as
       then the other acute angle must be labeled

Notice also that
       which is opposite over hypotenuse. Thus, when two angles are complementary, we can say that the sine of
       equals the cofunction of the complement of
       Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.
From these relationships, the cofunction identities are formed. Recall that you first encountered these identities in The Unit Circle: Sine and Cosine Functions.
Notice that the formulas in the table may also be justified algebraically using the sum and difference formulas. For example, using
we can write

### Using the Sum and Difference Formulas to Verify Identities

Verifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems. Reviewing the general rules presented earlier may help simplify the process of verifying an identity.

### Key Equations

### Key Concepts

1. The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles.
2. The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle. See  and .
3. The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the second angle. See .
4. The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions. See .
5. The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that the tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the product of the tangents of the angles. See .
6. The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles. See .
7. The cofunction identities apply to complementary angles and pairs of reciprocal functions. See .
8. Sum and difference formulas are useful in verifying identities. See  and .
9. Application problems are often easier to solve by using sum and difference formulas. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the exact value.
For the following exercises, rewrite in terms of
         and

For the following exercises, simplify the given expression.
For the following exercises, find the requested information.
For the following exercises, find the exact value of each expression.

### Graphical

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.
For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think

        )

### Technology

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

### Extensions

For the following exercises, prove the identities provided.
For the following exercises, prove or disprove the statements.


# Trigonometric Identities and Equations

## Double-Angle, Half-Angle, and Reduction Formulas

Bicycle and skateboard ramps made for competition (see ) must vary in height depending on the skill level of the competitors. For advanced competitors, the angle formed by the ramp and the ground should be
     such that
     The angle is divided in half for novices. What is the steepness of the ramp for novices? In this section, we will investigate three additional categories of identities that we can use to answer questions such as this one.

### Using Double-Angle Formulas to Find Exact Values

In the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another look at those same formulas. The double-angle formulas are a special case of the sum formulas, where
       Deriving the double-angle formula for sine begins with the sum formula,
If we let
       then we have
Deriving the double-angle for cosine gives us three options. First, starting from the sum formula,
       and letting
       we have
Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more variations. The first variation is:
The second variation is:
Similarly, to derive the double-angle formula for tangent, replacing
       in the sum formula gives

### Using Double-Angle Formulas to Verify Identities

Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.

### Use Reduction Formulas to Simplify an Expression

The double-angle formulas can be used to derive the reduction formulas, which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses, calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from the double-angle formulas.
We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Let’s begin with
       Solve for

Next, we use the formula
       Solve for

The last reduction formula is derived by writing tangent in terms of sine and cosine:

###  Using Half-Angle Formulas to Find Exact Values

The next set of identities is the set of half-angle formulas, which can be derived from the reduction formulas and we can use when we have an angle that is half the size of a special angle. If we replace
       with
       the half-angle formula for sine is found by simplifying the equation and solving for
       Note that the half-angle formulas are preceded by a
       sign. This does not mean that both the positive and negative expressions are valid. Rather, it depends on the quadrant in which
       terminates.
The half-angle formula for sine is derived as follows:
To derive the half-angle formula for cosine, we have
For the tangent identity, we have

### Key Equations

### Key Concepts

1. Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. See , ,  , and .
2. Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. See  and .
3. Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not. See , , and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the exact values of a)
          b)
          and c)
         without solving for

For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.
For the following exercises, simplify to one trigonometric expression.
For the following exercises, find the exact value using half-angle formulas.
For the following exercises, find the exact values of a)
          b)
          and c)
         without solving for
         when

For the following exercises, use  to find the requested half and double angles.
For the following exercises, simplify each expression. Do not evaluate.
For the following exercises, prove the given identity.
For the following exercises, rewrite the expression with an exponent no higher than 1.

### Technology

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.
For the following exercises, algebraically find an equivalent function, only in terms of
         and/or
         and then check the answer by graphing both functions.

### Extensions

For the following exercises, prove the identities.


# Trigonometric Identities and Equations

## Sum-to-Product and Product-to-Sum Formulas

A band marches down the field creating an amazing sound that bolsters the crowd. That sound travels as a wave that can be interpreted using trigonometric functions. For example,  represents a sound wave for the musical note A. In this section, we will investigate trigonometric identities that are the foundation of everyday phenomena such as sound waves.

### Expressing Products as Sums

We have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum. We can use the product-to-sum formulas, which express products of trigonometric functions as sums. Let’s investigate the cosine identity first and then the sine identity.

### Expressing Products as Sums for Cosine

We can derive the product-to-sum formula from the sum and difference identities for cosine. If we add the two equations, we get:
Then, we divide by
         to isolate the product of cosines:

### Expressing the Product of Sine and Cosine as a Sum

Next, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for sine. If we add the sum and difference identities, we get:
Then, we divide by 2 to isolate the product of cosine and sine:

### Expressing Products of Sines in Terms of Cosine

Expressing the product of sines in terms of cosine is also derived from the sum and difference identities for cosine. In this case, we will first subtract the two cosine formulas:
Then, we divide by 2 to isolate the product of sines:
Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.

### Expressing Sums as Products

Some problems require the reverse of the process we just used. The sum-to-product formulas allow us to express sums of sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few substitutions, we can derive the sum-to-product identity for sine. Let
       and

Then,
Thus, replacing
       and
       in the product-to-sum formula with the substitute expressions, we have
The other sum-to-product identities are derived similarly.

### Key Equations

### Key Concepts

1. From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine.
2. We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. See , , and .
3. We can also derive the sum-to-product identities from the product-to-sum identities using substitution.
4. We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines. See .
5. Trigonometric expressions are often simpler to evaluate using the formulas. See .
6. The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the other side. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, rewrite the product as a sum or difference.
For the following exercises, rewrite the sum or difference as a product.
For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.
For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.
For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.
For the following exercises, prove the identity.

### Numeric

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.

### Technology

For the following exercises, algebraically determine whether each of the given equation is an identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.
For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

### Extensions

For the following exercises, prove the following sum-to-product formulas.
For the following exercises, prove the identity.


# Trigonometric Identities and Equations

## Solving Trigonometric Equations

Thales of Miletus (circa 625–547 BC) is known as the founder of geometry. The legend is that he calculated the height of the Great Pyramid of Giza in Egypt using the theory of similar triangles, which he developed by measuring the shadow of his staff. He reasoned that when the height of his staff's shadow was exactly equal to the actual height of the staff, then the height of the nearby pyramid's shadow must also be equal to the height of the actual pyramid. Since the structures and their shadows were creating a right triangle with two equal sides, they were similar triangles. By measuring the length of the pyramid's shadow at that moment, he could obtain the height of the pyramid. Based on proportions, this theory has applications in a number of areas, including fractal geometry, engineering, and architecture. Often, the angle of elevation and the angle of depression are found using similar triangles.
In earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of the variable. In this section, we begin our study of trigonometric equations to study real-world scenarios such as the finding the dimensions of the pyramids.

### Solving Linear Trigonometric Equations in Sine and Cosine

Trigonometric equations are, as the name implies, equations that involve trigonometric functions. Similar in many ways to solving polynomial equations or rational equations, only specific values of the variable will be solutions, if there are solutions at all. Often we will solve a trigonometric equation over a specified interval. However, just as often, we will be asked to find all possible solutions, and as trigonometric functions are periodic, solutions are repeated within each period. In other words, trigonometric equations may have an infinite number of solutions. Additionally, like rational equations, the domain of the function must be considered before we assume that any solution is valid. The period of both the sine function and the cosine function is
   In other words, every
  units, the y-values repeat. If we need to find all possible solutions, then we must add
  where
  is an integer, to the initial solution. Recall the rule that gives the format for stating all possible solutions for a function where the period is

There are similar rules for indicating all possible solutions for the other trigonometric functions. Solving trigonometric equations requires the same techniques as solving algebraic equations. We read the equation from left to right, horizontally, like a sentence. We look for known patterns, factor, find common denominators, and substitute certain expressions with a variable to make solving a more straightforward process. However, with trigonometric equations, we also have the advantage of using the identities we developed in the previous sections.

### Solving Equations Involving a Single Trigonometric Function

When we are given equations that involve only one of the six trigonometric functions, their solutions involve using algebraic techniques and the unit circle (see ). We need to make several considerations when the equation involves trigonometric functions other than sine and cosine. Problems involving the reciprocals of the primary trigonometric functions need to be viewed from an algebraic perspective. In other words, we will write the reciprocal function, and solve for the angles using the function. Also, an equation involving the tangent function is slightly different from one containing a sine or cosine function. First, as we know, the period of tangent is
  not
  Further, the domain of tangent is all real numbers with the exception of odd integer multiples of
  unless, of course, a problem places its own restrictions on the domain.

### Solve Trigonometric Equations Using a Calculator

Not all functions can be solved exactly using only the unit circle. When we must solve an equation involving an angle other than one of the special angles, we will need to use a calculator. Make sure it is set to the proper mode, either degrees or radians, depending on the criteria of the given problem.

### Solving Trigonometric Equations in Quadratic Form

Solving a quadratic equation may be more complicated, but once again, we can use algebra as we would for any quadratic equation. Look at the pattern of the equation. Is there more than one trigonometric function in the equation, or is there only one? Which trigonometric function is squared? If there is only one function represented and one of the terms is squared, think about the standard form of a quadratic. Replace the trigonometric function with a variable such as
  or
  If substitution makes the equation look like a quadratic equation, then we can use the same methods for solving quadratics to solve the trigonometric equations.

### Solving Trigonometric Equations Using Fundamental Identities

While algebra can be used to solve a number of trigonometric equations, we can also use the fundamental identities because they make solving equations simpler. Remember that the techniques we use for solving are not the same as those for verifying identities. The basic rules of algebra apply here, as opposed to rewriting one side of the identity to match the other side. In the next example, we use two identities to simplify the equation.

### Solving Trigonometric Equations with Multiple Angles

Sometimes it is not possible to solve a trigonometric equation with identities that have a multiple angle, such as
  or
  When confronted with these equations, recall that
  is a horizontal compression by a factor of 2 of the function
  On an interval of
  we can graph two periods of
  as opposed to one cycle of
  This compression of the graph leads us to believe there may be twice as many x-intercepts or solutions to
  compared to
  This information will help us solve the equation.

### Solving Right Triangle Problems

We can now use all of the methods we have learned to solve problems that involve applying the properties of right triangles and the Pythagorean Theorem. We begin with the familiar Pythagorean Theorem,
  and model an equation to fit a situation.

### Key Concepts

1. When solving linear trigonometric equations, we can use algebraic techniques just as we do solving algebraic equations. Look for patterns, like the difference of squares, quadratic form, or an expression that lends itself well to substitution. See , , and .
2. Equations involving a single trigonometric function can be solved or verified using the unit circle. See , , and , and .
3. We can also solve trigonometric equations using a graphing calculator. See  and .
4. Many equations appear quadratic in form. We can use substitution to make the equation appear simpler, and then use the same techniques we use solving an algebraic quadratic: factoring, the quadratic formula, etc. See , , , and .
5. We can also use the identities to solve trigonometric equation. See , , and .
6. We can use substitution to solve a multiple-angle trigonometric equation, which is a compression of a standard trigonometric function. We will need to take the compression into account and verify that we have found all solutions on the given interval. See .
7. Real-world scenarios can be modeled and solved using the Pythagorean Theorem and trigonometric functions. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find all solutions exactly on the interval

For the following exercises, solve exactly on

For the following exercises, find all exact solutions on

For the following exercises, solve with the methods shown in this section exactly on the interval

For the following exercises, solve exactly on the interval
  Use the quadratic formula if the equations do not factor.
For the following exercises, find exact solutions on the interval
  Look for opportunities to use trigonometric identities.

### Graphical

For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.

### Technology

For the following exercises, use a calculator to find all solutions to four decimal places.
For the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval
  Round to four decimal places.

### Extensions

For the following exercises, find all solutions exactly to the equations on the interval

### Real-World Applications

For the following exercises, find a solution to the following word problem algebraically. Then use a calculator to verify the result. Round the answer to the nearest tenth of a degree.

### Chapter Review Exercises

### Solving Trigonometric Equations with Identities

For the following exercises, find all solutions exactly that exist on the interval

For the following exercises, use basic identities to simplify the expression.
For the following exercises, determine if the given identities are equivalent.

### Sum and Difference Identities

For the following exercises, find the exact value.
For the following exercises, prove the identity.
For the following exercise, simplify the expression.
For the following exercises, find the exact value.

### Double-Angle, Half-Angle, and Reduction Formulas

For the following exercises, find the exact value.
For the following exercises, use  to find the desired quantities.
For the following exercises, prove the identity.
For the following exercises, rewrite the expression with no powers.

### Sum-to-Product and Product-to-Sum Formulas

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.
For the following exercises, evaluate the sum by using a product formula. Write the exact answer.
For the following exercises, change the functions from a product to a sum or a sum to a product.

### Solving Trigonometric Equations

For the following exercises, find all exact solutions on the interval

For the following exercises, find all exact solutions on the interval

For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval
     Round to four decimal places.
For the following exercises, graph each side of the equation to find the approximate solutions on the interval

### Practice Test

For the following exercises, simplify the given expression.
For the following exercises, find the exact value.
For the following exercises, simplify each expression. Do not evaluate.
For the following exercises, find all exact solutions to the equation on

For the following exercise, rewrite the product as a sum or difference.
For the following exercise, rewrite the sum or difference as a product.
For the following exercises, find all solutions exactly on the interval

For the following exercises, prove the identity.


# Further Applications of Trigonometry

## Introduction to Further Applications of Trigonometry

The world’s largest tree by volume, named General Sherman, stands 274.9 feet tall and resides in Northern California.Source: National Park Service. "The General Sherman Tree." http://www.nps.gov/seki/naturescience/sherman.htm. Accessed April 25, 2014. Just how do scientists know its true height? A common way to measure the height involves determining the angle of elevation, which is formed by the tree and the ground at a point some distance away from the base of the tree. This method is much more practical than climbing the tree and dropping a very long tape measure.
In this chapter, we will explore applications of trigonometry that will enable us to solve many different kinds of problems, including finding the height of a tree. We extend topics we introduced in Trigonometric Functions and investigate applications more deeply and meaningfully.


# Further Applications of Trigonometry

## Non-right Triangles: Law of Sines

To ensure the safety of over 5,000 U.S. aircraft flying simultaneously during peak times, air traffic controllers monitor and communicate with them after receiving data from the robust radar beacon system. Suppose two radar stations located 20 miles apart each detect an aircraft between them. The angle of elevation measured by the first station is 35 degrees, whereas the angle of elevation measured by the second station is 15 degrees. How can we determine the altitude of the aircraft? We see in  that the triangle formed by the aircraft and the two stations is not a right triangle, so we cannot use what we know about right triangles. In this section, we will find out how to solve problems involving non-right triangles.

### Using the Law of Sines to Solve Oblique Triangles

In any triangle, we can draw an altitude, a perpendicular line from one vertex to the opposite side, forming two right triangles. It would be preferable, however, to have methods that we can apply directly to non-right triangles without first having to create right triangles.
Any triangle that is not a right triangle is an oblique triangle. Solving an oblique triangle means finding the measurements of all three angles and all three sides. To do so, we need to start with at least three of these values, including at least one of the sides. We will investigate three possible oblique triangle problem situations:
1. ASA (angle-side-angle) We know the measurements of two angles and the included side. See .

2. AAS (angle-angle-side) We know the measurements of two angles and a side that is not between the known angles. See .

3. SSA (side-side-angle) We know the measurements of two sides and an angle that is not between the known sides. See .

Knowing how to approach each of these situations enables us to solve oblique triangles without having to drop a perpendicular to form two right triangles. Instead, we can use the fact that the ratio of the measurement of one of the angles to the length of its opposite side will be equal to the other two ratios of angle measure to opposite side. Let’s see how this statement is derived by considering the triangle shown in .
Using the right triangle relationships, we know that
  and
  Solving both equations for
       gives two different expressions for

We then set the expressions equal to each other.
Similarly, we can compare the other ratios.
Collectively, these relationships are called the Law of Sines.
Note the standard way of labeling triangles: angle
       (alpha) is opposite side
       angle
       (beta) is opposite side
       and angle
       (gamma) is opposite side
       See .
While calculating angles and sides, be sure to carry the exact values through to the final answer. Generally, final answers are rounded to the nearest tenth, unless otherwise specified.

### Using The Law of Sines to Solve SSA Triangles

We can use the Law of Sines to solve any oblique triangle, but some solutions may not be straightforward. In some cases, more than one triangle may satisfy the given criteria, which we describe as an ambiguous case. Triangles classified as SSA, those in which we know the lengths of two sides and the measurement of the angle opposite one of the given sides, may result in one or two solutions, or even no solution.

###  Finding the Area of an Oblique Triangle Using the Sine Function

Now that we can solve a triangle for missing values, we can use some of those values and the sine function to find the area of an oblique triangle. Recall that the area formula for a triangle is given as
       where
       is base and
       is height. For oblique triangles, we must find
       before we can use the area formula. Observing the two triangles in , one acute and one obtuse, we can drop a perpendicular to represent the height and then apply the trigonometric property
  to write an equation for area in oblique triangles. In the acute triangle, we have
       or
       However, in the obtuse triangle, we drop the perpendicular outside the triangle and extend the base
       to form a right triangle. The angle used in calculation is
       or

Thus,
Similarly,

###  Solving Applied Problems Using the Law of Sines

The more we study trigonometric applications, the more we discover that the applications are countless. Some are flat, diagram-type situations, but many applications in calculus, engineering, and physics involve three dimensions and motion.

### Key Equations

### Key Concepts

1. The Law of Sines can be used to solve oblique triangles, which are non-right triangles.
2. According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side.
3. There are three possible cases: ASA, AAS, SSA. Depending on the information given, we can choose the appropriate equation to find the requested solution. See .
4. The ambiguous case arises when an oblique triangle can have different outcomes.
5. There are three possible cases that arise from SSA arrangement—a single solution, two possible solutions, and no solution. See  and .
6. The Law of Sines can be used to solve triangles with given criteria. See .
7. The general area formula for triangles translates to oblique triangles by first finding the appropriate height value. See .
8. There are many trigonometric applications. They can often be solved by first drawing a diagram of the given information and then using the appropriate equation. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, assume
     is opposite side
     is opposite side
     and
     is opposite side
     Solve each triangle, if possible. Round each answer to the nearest tenth.
For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle
     is opposite side
     angle
     is opposite side
     and angle
     is opposite side

For the following exercises, assume
     is opposite side
     is opposite side
     and
     is opposite side
     Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.
For the following exercises, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles in the ambiguous case. Round each answer to the nearest tenth.
For the following exercises, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.

### Graphical

For the following exercises, find the length of side
     Round to the nearest tenth.
For the following exercises, find the measure of angle
  if possible. Round to the nearest tenth.
For the following exercise, solve the triangle. Round each answer to the nearest tenth.

### Extensions

### Real-World Applications


# Further Applications of Trigonometry

## Non-right Triangles: Law of Cosines

Suppose a boat leaves port, travels 10 miles, turns 20 degrees, and travels another 8 miles as shown in . How far from port is the boat?
Unfortunately, while the Law of Sines enables us to address many non-right triangle cases, it does not help us with triangles where the known angle is between two known sides, a SAS (side-angle-side) triangle, or when all three sides are known, but no angles are known, a SSS (side-side-side) triangle. In this section, we will investigate another tool for solving oblique triangles described by these last two cases.

### Using the Law of Cosines to Solve Oblique Triangles

The tool we need to solve the problem of the boat’s distance from the port is the Law of Cosines, which defines the relationship among angle measurements and side lengths in oblique triangles. Three formulas make up the Law of Cosines. At first glance, the formulas may appear complicated because they include many variables. However, once the pattern is understood, the Law of Cosines is easier to work with than most formulas at this mathematical level.
Understanding how the Law of Cosines is derived will be helpful in using the formulas. The derivation begins with the Generalized Pythagorean Theorem, which is an extension of the Pythagorean Theorem to non-right triangles. Here is how it works: An arbitrary non-right triangle
       is placed in the coordinate plane with vertex
       at the origin, side
       drawn along the x-axis, and vertex
       located at some point
       in the plane, as illustrated in . Generally, triangles exist anywhere in the plane, but for this explanation we will place the triangle as noted.
We can drop a perpendicular from
       to the x-axis (this is the altitude or height). Recalling the basic trigonometric identities, we know that
In terms of
  and

 The
       point located at
       has coordinates

           Using the side
       as one leg of a right triangle and
       as the second leg, we can find the length of hypotenuse
       using the Pythagorean Theorem. Thus,
The formula derived is one of the three equations of the Law of Cosines. The other equations are found in a similar fashion.
Keep in mind that it is always helpful to sketch the triangle when solving for angles or sides. In a real-world scenario, try to draw a diagram of the situation. As more information emerges, the diagram may have to be altered. Make those alterations to the diagram and, in the end, the problem will be easier to solve.

### Solving Applied Problems Using the Law of Cosines

Just as the Law of Sines provided the appropriate equations to solve a number of applications, the Law of Cosines is applicable to situations in which the given data fits the cosine models. We may see these in the fields of navigation, surveying, astronomy, and geometry, just to name a few.

### Using Heron’s Formula to Find the Area of a Triangle

We already learned how to find the area of an oblique triangle when we know two sides and an angle. We also know the formula to find the area of a triangle using the base and the height. When we know the three sides, however, we can use Heron’s formula instead of finding the height. Heron of Alexandria was a geometer who lived during the first century A.D. He discovered a formula for finding the area of oblique triangles when three sides are known.

### Key Equations

### Key Concepts

1. The Law of Cosines defines the relationship among angle measurements and lengths of sides in oblique triangles.
2. The Generalized Pythagorean Theorem is the Law of Cosines for two cases of oblique triangles: SAS and SSS. Dropping an imaginary perpendicular splits the oblique triangle into two right triangles or forms one right triangle, which allows sides to be related and measurements to be calculated. See  and .
3. The Law of Cosines is useful for many types of applied problems. The first step in solving such problems is generally to draw a sketch of the problem presented. If the information given fits one of the three models (the three equations), then apply the Law of Cosines to find a solution. See  and .
4. Heron’s formula allows the calculation of area in oblique triangles. All three sides must be known to apply Heron’s formula. See  and See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, assume
         is opposite side
          is opposite side
         and
          is opposite side
         If possible, solve each triangle for the unknown side. Round to the nearest tenth.
For the following exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.
For the following exercises, solve the triangle. Round to the nearest tenth.
For the following exercises, use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

### Graphical

For the following exercises, find the length of side
   Round to the nearest tenth.
For the following exercises, find the measurement of angle

For the following exercises, solve for the unknown side. Round to the nearest tenth.
For the following exercises, find the area of the triangle. Round to the nearest hundredth.

### Extensions

For the following exercises, suppose that
         represents the relationship of three sides of a triangle and the cosine of an angle.
For the following exercises, find the area of the triangle.

### Real-World Applications


# Further Applications of Trigonometry

## Polar Coordinates

Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (see ). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of representing location that is different from a standard coordinate grid.

### Plotting Points Using Polar Coordinates

When we think about plotting points in the plane, we usually think of rectangular coordinates
  in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeled
  and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.
The polar grid is scaled as the unit circle with the positive x-axis now viewed as the polar axis and the origin as the pole. The first coordinate
  is the radius or length of the directed line segment from the pole. The angle
   measured in radians, indicates the direction of
  We move counterclockwise from the polar axis by an angle of
  and measure a directed line segment the length of
  in the direction of
  Even though we measure
  first and then
   the polar point is written with the r-coordinate first. For example, to plot the point
  we would move
  units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in .

### Converting from Polar Coordinates to Rectangular Coordinates

When given a set of polar coordinates, we may need to convert them to rectangular coordinates. To do so, we can recall the relationships that exist among the variables
  and

Dropping a perpendicular from the point in the plane to the x-axis forms a right triangle, as illustrated in . An easy way to remember the equations above is to think of
  as the adjacent side over the hypotenuse and
  as the opposite side over the hypotenuse.

### Converting from Rectangular Coordinates to Polar Coordinates

To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.

### Transforming Equations between Polar and Rectangular Forms

We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.

###  Identify and Graph Polar Equations by Converting to Rectangular Equations

We have learned how to convert rectangular coordinates to polar coordinates, and we have seen that the points are indeed the same. We have also transformed polar equations to rectangular equations and vice versa. Now we will demonstrate that their graphs, while drawn on different grids, are identical.

### Key Equations

### Key Concepts

1. The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.
2. To plot a point in the form
  move in a counterclockwise direction from the polar axis by an angle of
  and then extend a directed line segment from the pole the length of
   in the direction of
  If
  is negative, move in a clockwise direction, and extend a directed line segment the length of
   in the direction of
   See .
3. If
  is negative, extend the directed line segment in the opposite direction of
  See .
4. To convert from polar coordinates to rectangular coordinates, use the formulas
  and
  See  and .
5. To convert from rectangular coordinates to polar coordinates, use one or more of the formulas:
  and
  See .
6. Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations. See , , and .
7. Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and then graph it in the rectangular plane. See , , and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, convert the given polar coordinates to Cartesian coordinates. Remember to consider the quadrant in which the given point is located when determining
  for the point.
For the following exercises, convert the given Cartesian coordinates to polar coordinates with
  Remember to consider the quadrant in which the given point is located.
For the following exercises, convert the given Cartesian equation to a polar equation.
For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

### Graphical

For the following exercises, find the polar coordinates of the point.
For the following exercises, plot the points.
For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

### Technology

### Extensions

For the following exercise, graph the polar inequality.


# Further Applications of Trigonometry

## Polar Coordinates: Graphs

The planets move through space in elliptical, periodic orbits about the sun, as shown in . They are in constant motion, so fixing an exact position of any planet is valid only for a moment. In other words, we can fix only a planet’s instantaneous position. This is one application of polar coordinates, represented as

 We interpret
  as the distance from the center of the sun and
  as the planet’s angular bearing, or its direction from the center of the sun. In this section, we will focus on the polar system and the graphs that are generated directly from polar coordinates.

### Testing Polar Equations for Symmetry

Just as a rectangular equation such as
  describes the relationship between
  and
  on a Cartesian grid, a polar equation describes a relationship between
  and
  on a polar grid. Recall that the coordinate pair
  indicates that we move counterclockwise from the polar axis (positive x-axis) by an angle of
  and extend a ray from the pole (origin)
  units in the direction of
  All points that satisfy the polar equation are on the graph.
Symmetry is a property that helps us recognize and plot the graph of any equation. If an equation has a graph that is symmetric with respect to an axis, it means that if we folded the graph in half over that axis, the portion of the graph on one side would coincide with the portion on the other side. By performing three tests, we will see how to apply the properties of symmetry to polar equations. Further, we will use symmetry (in addition to plotting key points, zeros, and maximums of

to determine the graph of a polar equation.
In the first test, we consider symmetry with respect to the line
  (y-axis). We replace
  with
  to determine if the new equation is equivalent to the original equation. For example, suppose we are given the equation

This equation exhibits symmetry with respect to the line

In the second test, we consider symmetry with respect to the polar axis (
 -axis). We replace
  with
  or
  to determine equivalency between the tested equation and the original. For example, suppose we are given the equation

The graph of this equation exhibits symmetry with respect to the polar axis.
In the third test, we consider symmetry with respect to the pole (origin). We replace
  with
  to determine if the tested equation is equivalent to the original equation. For example, suppose we are given the equation

The equation has failed the symmetry test, but that does not mean that it is not symmetric with respect to the pole. Passing one or more of the symmetry tests verifies that symmetry will be exhibited in a graph. However, failing the symmetry tests does not necessarily indicate that a graph will not be symmetric about the line
  the polar axis, or the pole. In these instances, we can confirm that symmetry exists by plotting reflecting points across the apparent axis of symmetry or the pole. Testing for symmetry is a technique that simplifies the graphing of polar equations, but its application is not perfect.

### Graphing Polar Equations by Plotting Points

To graph in the rectangular coordinate system we construct a table of
  and
  values. To graph in the polar coordinate system we construct a table of
  and
  values. We enter values of

into a polar equation and calculate
  However, using the properties of symmetry and finding key values of
  and
  means fewer calculations will be needed.

### Finding Zeros and Maxima

To find the zeros of a polar equation, we solve for the values of
  that result in
   Recall that, to find the zeros of polynomial functions, we set the equation equal to zero and then solve for
  We use the same process for polar equations. Set
  and solve for

For many of the forms we will encounter, the maximum value of a polar equation is found by substituting those values of
  into the equation that result in the maximum value of the trigonometric functions. Consider
  the maximum distance between the curve and the pole is 5 units.  The maximum value of the cosine function is 1 when
  so our polar equation is
  and the value
   will yield the maximum

Similarly, the maximum value of the sine function is 1 when
  and if our polar equation is
  the value
  will yield the maximum
  We may find additional information by calculating values of
  when
  These points would be polar axis intercepts, which may be helpful in drawing the graph and identifying the curve of a polar equation.

### Investigating Circles

Now we have seen the equation of a circle in the polar coordinate system. In the last two examples, the same equation was used to illustrate the properties of symmetry and demonstrate how to find the zeros, maximum values, and plotted points that produced the graphs. However, the circle is only one of many shapes in the set of polar curves.
There are five classic polar curves: cardioids, limaҫons, lemniscates, rose curves, and Archimedes’ spirals. We will briefly touch on the polar formulas for the circle before moving on to the classic curves and their variations.

### Investigating Cardioids

While translating from polar coordinates to Cartesian coordinates may seem simpler in some instances, graphing the classic curves is actually less complicated in the polar system. The next curve is called a cardioid, as it resembles a heart. This shape is often included with the family of curves called limaçons, but here we will discuss the cardioid on its own.

### Investigating Limaçons

The word limaçon is Old French for “snail,” a name that describes the shape of the graph. As mentioned earlier, the cardioid is a member of the limaçon family, and we can see the similarities in the graphs. The other images in this category include the one-loop limaçon and the two-loop (or inner-loop) limaçon. One-loop limaçons are sometimes referred to as dimpled limaçons when
  and convex limaçons when

Another type of limaçon, the inner-loop limaçon, is named for the loop formed inside the general limaçon shape. It was discovered by the German artist Albrecht Dürer(1471-1528), who revealed a method for drawing the inner-loop limaçon in his 1525 book Underweysung der Messing. A century later, the father of mathematician Blaise Pascal, Étienne Pascal(1588-1651), rediscovered it.

### Investigating Lemniscates

The lemniscate is a polar curve resembling the infinity symbol
  or a figure 8. Centered at the pole, a lemniscate is symmetrical by definition.

###  Investigating Rose Curves

The next type of polar equation produces a petal-like shape called a rose curve. Although the graphs look complex, a simple polar equation generates the pattern.

### Investigating the Archimedes’ Spiral

The final polar equation we will discuss is the Archimedes’ spiral, named for its discoverer, the Greek mathematician Archimedes (c. 287 BCE-c. 212 BCE), who is credited with numerous discoveries in the fields of geometry and mechanics.

### Summary of Curves

We have explored a number of seemingly complex polar curves in this section.  and  summarize the graphs and equations for each of these curves.

### Key Concepts

1. It is easier to graph polar equations if we can test the equations for symmetry with respect to the line
  the polar axis, or the pole.
2. There are three symmetry tests that indicate whether the graph of a polar equation will exhibit symmetry. If an equation fails a symmetry test, the graph may or may not exhibit symmetry. See .
3. Polar equations may be graphed by making a table of values for
  and

4. The maximum value of a polar equation is found by substituting the value
  that leads to the maximum value of the trigonometric expression.
5. The zeros of a polar equation are found by setting
  and solving for
  See .
6. Some formulas that produce the graph of a circle in polar coordinates are given by
  and
  See .
7. The formulas that produce the graphs of a cardioid are given by
  and
  for
   and
  See .
8. The formulas that produce the graphs of a one-loop limaçon are given by
  and
  for
  See .
9. The formulas that produce the graphs of an inner-loop limaçon are given by
  and
  for

and

See .
10. The formulas that produce the graphs of a lemniscates are given by
  and
  where
  See .
11. The formulas that produce the graphs of rose curves are given by
  and
  where
  if
  is even, there are
  petals, and if
  is odd, there are
  petals. See  and .
12. The formula that produces the graph of an Archimedes’ spiral is given by
   See .

### Section Exercises

### Verbal

### Graphical

For the following exercises, test the equation for symmetry.
For the following exercises, graph the polar equation. Identify the name of the shape.

### Technology

For the following exercises, use a graphing calculator to sketch the graph of the polar equation.
For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of
   and then explain the differences shown in the graphs.

### Extensions

For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.


# Further Applications of Trigonometry

## Polar Form of Complex Numbers

“God made the integers; all else is the work of man.” This rather famous quote by nineteenth-century German mathematician Leopold Kronecker sets the stage for this section on the polar form of a complex number. Complex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as Pythagoras, Descartes, De Moivre, Euler, Gauss, and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science.
We first encountered complex numbers in Complex Numbers. In this section, we will focus on the mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De Moivre’s Theorem.

### Plotting Complex Numbers in the Complex Plane

Plotting a complex number
  is similar to plotting a real number, except that the horizontal axis represents the real part of the number,
  and the vertical axis represents the imaginary part of the number,

### Finding the Absolute Value of a Complex Number

The first step toward working with a complex number in polar form is to find the absolute value. The absolute value of a complex number is the same as its magnitude, or
  It measures the distance from the origin to a point in the plane. For example, the graph of
  in , shows

### Writing Complex Numbers in Polar Form

The polar form of a complex number expresses a number in terms of an angle
  and its distance from the origin
  Given a complex number in rectangular form expressed as
  we use the same conversion formulas as we do to write the number in trigonometric form:
We review these relationships in .
We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the point
  The modulus, then, is the same as
  the radius in polar form. We use
  to indicate the angle of direction (just as with polar coordinates). Substituting, we have

### Converting a Complex Number from Polar to Rectangular Form

Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, given
  first evaluate the trigonometric functions
  and
  Then, multiply through by

### Finding Products of Complex Numbers in Polar Form

Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. For the rest of this section, we will work with formulas developed by French mathematician Abraham De Moivre (1667-1754). These formulas have made working with products, quotients, powers, and roots of complex numbers much simpler than they appear. The rules are based on multiplying the moduli and adding the arguments.

### Finding Quotients of Complex Numbers in Polar Form

The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments.

### Finding Powers of Complex Numbers in Polar Form

Finding powers of complex numbers is greatly simplified using De Moivre’s Theorem. It states that, for a positive integer
  is found by raising the modulus to the
  power and multiplying the argument by
  It is the standard method used in modern mathematics.

### Finding Roots of Complex Numbers in Polar Form

To find the  in polar form, we use the
  Root Theorem or De Moivre’s Theorem and raise the complex number to a power with a rational exponent. There are several ways to represent a formula for finding   roots of complex numbers in polar form.

### Key Concepts

1. Complex numbers in the form
  are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Label the x-axis as the real axis and the y-axis as the imaginary axis. See .
2. The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point:
  See  and .
3. To write complex numbers in polar form, we use the formulas
  and
  Then,
  See  and .
4. To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through by
  See  and .
5. To find the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate the trigonometric functions, and multiply using the distributive property. See .
6. To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference of the two angles. See .
7. To find the power of a complex number
  raise
   to the power
   and multiply
   by
  See .
8. Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the absolute value of the given complex number.
For the following exercises, write the complex number in polar form.
For the following exercises, convert the complex number from polar to rectangular form.
For the following exercises, find
  in polar form.
For the following exercises, find
  in polar form.
For the following exercises, find the powers of each complex number in polar form.
For the following exercises, evaluate each root.

### Graphical

For the following exercises, plot the complex number in the complex plane.

### Technology

For the following exercises, find all answers rounded to the nearest hundredth.