# 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. 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 word problem algebraically. Then use a calculator to verify the result. Round the answer to the nearest tenth of a degree.


# Trigonometric Identities and Equations

## Modeling with Trigonometric Functions

Suppose we charted the average daily temperatures in New York City over the course of one year. We would expect to find the lowest temperatures in January and February and highest in July and August. This familiar cycle repeats year after year, and if we were to extend the graph over multiple years, it would resemble a periodic function.
Many other natural phenomena are also periodic. For example, the phases of the moon have a period of approximately 28 days, and birds know to fly south at about the same time each year.
So how can we model an equation to reflect periodic behavior? First, we must collect and record data. We then find a function that resembles an observed pattern. Finally, we make the necessary alterations to the function to get a model that is dependable. In this section, we will take a deeper look at specific types of periodic behavior and model equations to fit data.

### Determining the Amplitude and Period of a Sinusoidal Function

Any motion that repeats itself in a fixed time period is considered periodic motion and can be modeled by a sinusoidal function. The amplitude of a sinusoidal function is the distance from the midline to the maximum value, or from the midline to the minimum value. The midline is the average value. Sinusoidal functions oscillate above and below the midline, are periodic, and repeat values in set cycles. Recall from Graphs of the Sine and Cosine Functions that the period of the sine function and the cosine function is
  In other words, for any value of

### Finding Equations and Graphing Sinusoidal Functions

One method of graphing sinusoidal functions is to find five key points. These points will correspond to intervals of equal length representing
  of the period. The key points will indicate the location of maximum and minimum values. If there is no vertical shift, they will also indicate x-intercepts. For example, suppose we want to graph the function
  We know that the period is
  so we find the interval between key points as follows.
Starting with
  we calculate the first y-value, add the length of the interval
  to 0, and calculate the second y-value. We then add
  repeatedly until the five key points are determined. The last value should equal the first value, as the calculations cover one full period. Making a table similar to , we can see these key points clearly on the graph shown in .

### Modeling Periodic Behavior

We will now apply these ideas to problems involving periodic behavior.

### Modeling Harmonic Motion Functions

Harmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general periodic motion applications cycle through their periods with no outside interference, harmonic motion requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.

### Simple Harmonic Motion

A type of motion described as simple harmonic motion involves a restoring force but assumes that the motion will continue forever. Imagine a weighted object hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of the object from its equilibrium point. When

### Damped Harmonic Motion

In reality, a pendulum does not swing back and forth forever, nor does an object on a spring bounce up and down forever. Eventually, the pendulum stops swinging and the object stops bouncing and both return to equilibrium. Periodic motion in which an energy-dissipating force, or damping factor, acts is known as damped harmonic motion. Friction is typically the damping factor.
In physics, various formulas are used to account for the damping factor on the moving object. Some of these are calculus-based formulas that involve derivatives. For our purposes, we will use formulas for basic damped harmonic motion models.

### Bounding Curves in Harmonic Motion

Harmonic motion graphs may be enclosed by bounding curves. When a function has a varying amplitude, such that the amplitude rises and falls multiple times within a period, we can determine the bounding curves from part of the function.

### Key Equations

### Key Concepts

1. Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts. See  and .
2. Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values. See .
3. Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year. See , ,  and .
4. Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight. See .
5. Damped harmonic motion is a form of periodic behavior affected by a damping factor. Energy dissipating factors, like friction, cause the displacement of the object to shrink. See , , , , and .
6. Bounding curves delineate the graph of harmonic motion with variable maximum and minimum values. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.

### Graphical

For the following exercises, graph the given function, and then find a possible physical process that the equation could model.

### Technology

For the following exercise, construct a function modeling behavior and use a calculator to find desired results.

### Real-World Applications

For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.
For the following exercises, find the amplitude, period, and frequency of the given function.
For the following exercises, construct an equation that models the described behavior.
For the following exercises, construct an equation that models the described behavior.
For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.

### Extensions

For the following exercises, find a function of the form
  that fits the given data.
For the following exercises, find a function of the form
   that fits the given data.

### 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 zeroes on the interval

### Modeling with Trigonometric Equations

For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table.
For the following exercises, construct functions that model the described behavior.
For the following exercises, find the amplitude, frequency, and period of the given equations.
For the following exercises, model the described behavior and find requested values.

### Practice Test

For the following exercises, simplify the given expression.
For the following exercises, find the exact value.
For the following exercises, find all exact solutions to the equation on

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.


# Further Applications of Trigonometry

## Parametric Equations

Consider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen in . At any moment, the moon is located at a particular spot relative to the planet. But how do we write and solve the equation for the position of the moon when the distance from the planet, the speed of the moon’s orbit around the planet, and the speed of rotation around the sun are all unknowns? We can solve only for one variable at a time.
In this section, we will consider sets of equations given by
   and
   where
   is the independent variable of time. We can use these parametric equations in a number of applications when we are looking for not only a particular position but also the direction of the movement. As we trace out successive values of
  the orientation of the curve becomes clear. This is one of the primary advantages of using parametric equations: we are able to trace the movement of an object along a path according to time. We begin this section with a look at the basic components of parametric equations and what it means to parameterize a curve. Then we will learn how to eliminate the parameter, translate the equations of a curve defined parametrically into rectangular equations, and find the parametric equations for curves defined by rectangular equations.

### Parameterizing a Curve

When an object moves along a curve—or curvilinear path—in a given direction and in a given amount of time, the position of the object in the plane is given by the x-coordinate and the y-coordinate. However, both
  and

vary over time and so are functions of time. For this reason, we add another variable, the parameter, upon which both
  and
  are dependent functions. In the example in the section opener, the parameter is time,
  The
  position of the moon at time,
  is represented as the function
  and the
  position of the moon at time,
  is represented as the function
  Together,
   and
   are called parametric equations, and generate an ordered pair
  Parametric equations primarily describe motion and direction.
When we parameterize a curve, we are translating a single equation in two variables, such as
  and
  into an equivalent pair of equations in three variables,
  and
  One of the reasons we parameterize a curve is because the parametric equations yield more information: specifically, the direction of the object’s motion over time.
When we graph parametric equations, we can observe the individual behaviors of
  and of
  There are a number of shapes that cannot be represented in the form
  meaning that they are not functions. For example, consider the graph of a circle, given as
  Solving for
  gives
  or two equations:
  and
  If we graph
  and
  together, the graph will not pass the vertical line test, as shown in . Thus, the equation for the graph of a circle is not a function.
However, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.

### Eliminating the Parameter

In many cases, we may have a pair of parametric equations but find that it is simpler to draw a curve if the equation involves only two variables, such as
  and
  Eliminating the parameter is a method that may make graphing some curves easier. However, if we are concerned with the mapping of the equation according to time, then it will be necessary to indicate the orientation of the curve as well. There are various methods for eliminating the parameter
  from a set of parametric equations; not every method works for every type of equation. Here we will review the methods for the most common types of equations.

### Eliminating the Parameter from Polynomial, Exponential, and Logarithmic Equations

For polynomial, exponential, or logarithmic equations expressed as two parametric equations, we choose the equation that is most easily manipulated and solve for
  We substitute the resulting expression for

into the second equation. This gives one equation in
  and

### Eliminating the Parameter from Trigonometric Equations

Eliminating the parameter from trigonometric equations is a straightforward substitution. We can use a few of the familiar trigonometric identities and the Pythagorean Theorem.
First, we use the identities:
Solving for
  and
  we have
Then, use the Pythagorean Theorem:
Substituting gives

### Finding Cartesian Equations from Curves Defined Parametrically

When we are given a set of parametric equations and need to find an equivalent Cartesian equation, we are essentially “eliminating the parameter.” However, there are various methods we can use to rewrite a set of parametric equations as a Cartesian equation. The simplest method is to set one equation equal to the parameter, such as
  In this case,
  can be any expression. For example, consider the following pair of equations.
Rewriting this set of parametric equations is a matter of substituting
  for
  Thus, the Cartesian equation is

### Finding Parametric Equations for Curves Defined by Rectangular Equations

Although we have just shown that there is only one way to interpret a set of parametric equations as a rectangular equation, there are multiple ways to interpret a rectangular equation as a set of parametric equations. Any strategy we may use to find the parametric equations is valid if it produces equivalency. In other words, if we choose an expression to represent
  and then substitute it into the
  equation, and it produces the same graph over the same domain as the rectangular equation, then the set of parametric equations is valid. If the domain becomes restricted in the set of parametric equations, and the function does not allow the same values for
  as the domain of the rectangular equation, then the graphs will be different.

### Key Concepts

1. Parameterizing a curve involves translating a rectangular equation in two variables,
  and
  into two equations in three variables, x, y, and t. Often, more information is obtained from a set of parametric equations. See , , and .
2. Sometimes equations are simpler to graph when written in rectangular form. By eliminating
  an equation in
  and
  is the result.
3. To eliminate
  solve one of the equations for
  and substitute the expression into the second equation. See , , , and .
4. Finding the rectangular equation for a curve defined parametrically is basically the same as eliminating the parameter. Solve for
  in one of the equations, and substitute the expression into the second equation. See .
5. There are an infinite number of ways to choose a set of parametric equations for a curve defined as a rectangular equation.
6. Find an expression for
  such that the domain of the set of parametric equations remains the same as the original rectangular equation. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, eliminate the parameter
  to rewrite the parametric equation as a Cartesian equation.
For the following exercises, rewrite the parametric equation as a Cartesian equation by building an
   table.
For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting
   or by setting

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using
   and
  Identify the curve.

### Technology

For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.
For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.

### Extensions


# Further Applications of Trigonometry

## Parametric Equations: Graphs

While not every fan (or team manager) appreciates it, baseball and many other sports have become dependent on analytics, which involve complex data recording and quantitative evaluation used to understand and predict behavior. The earliest influence of analytics was mostly statistical; more recently, physics and other sciences have come into play. Foremost among these is the focus on launch angle and exit velocity, which when at certain values can almost guarantee a home run. On the other hand, emphasis on launch angle and focusing on home runs rather than overall hitting results in far more outs. Consider the following situation: it is the bottom of the ninth inning, with two outs and two players on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately
  to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using parametric equations. In this section, we’ll discuss parametric equations and some common applications, such as projectile motion problems.

### Graphing Parametric Equations by Plotting Points

In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.

### Applications of Parametric Equations

Many of the advantages of parametric equations become obvious when applied to solving real-world problems. Although rectangular equations in x and y give an overall picture of an object's path, they do not reveal the position of an object at a specific time. Parametric equations, however, illustrate how the values of x and y change depending on t, as the location of a moving object at a particular time.
A common application of parametric equations is solving problems involving projectile motion. In this type of motion, an object is propelled forward in an upward direction forming an angle of
            to the horizontal, with an initial speed of
           and at a height
            above the horizontal.
The path of an object propelled at an inclination of
              to the horizontal, with initial speed
  and at a height
              above the horizontal, is given by
where
             accounts for the effects of gravity and
              is the initial height of the object. Depending on the units involved in the problem, use
  or
  The equation for
             gives horizontal distance, and the equation for
              gives the vertical distance.

### Key Concepts

1. When there is a third variable, a third parameter on which
           and
           depend, parametric equations can be used.
2. To graph parametric equations by plotting points, make a table with three columns labeled
           and
           Choose values for
           in increasing order. Plot the last two columns for
           and
           See  and .
3. When graphing a parametric curve by plotting points, note the associated t-values and show arrows on the graph indicating the orientation of the curve. See  and .
4. Parametric equations allow the direction or the orientation of the curve to be shown on the graph. Equations that are not functions can be graphed and used in many applications involving motion. See .
5. Projectile motion depends on two parametric equations:
  and
  Initial velocity is symbolized as
   represents the initial angle of the object when thrown, and
            represents the height at which the object is propelled.

### Section Exercises

### Verbal

### Graphical

For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.
For the following exercises, sketch the curve and include the orientation.
For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.
For the following exercises, graph the equation and include the orientation.
For the following exercises, use the parametric equations for integers a and b:
For the following exercises, describe the graph of the set of parametric equations.
For the following exercises, use a graphing utility to graph on the window
     by
     on the domain
     for the following values of
     and
    , and include the orientation.

### Technology

For the following exercises, look at the graphs that were created by parametric equations of the form
  Use the parametric mode on the graphing calculator to find the values of
              and
              to achieve each graph.
For the following exercises, use a graphing utility to graph the given parametric equations.

### Extensions

For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 64 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time
             Neglect air resistance.
For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain.


# Further Applications of Trigonometry

## Vectors

An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour, as shown in . What are the ground speed and actual bearing of the plane?
Ground speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane’s groundspeed and bearing, while investigating another approach to problems of this type. First, however, let’s examine the basics of vectors.

### A Geometric View of Vectors

A vector is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point, where it begins, and a terminal point, where it ends. A vector is defined by its magnitude, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:
1. Lower case, boldfaced type, with or without an arrow on top such as

2. Given initial point
  and terminal point
  a vector can be represented as
  The arrowhead on top is what indicates that it is not just a line, but a directed line segment.
3. Given an initial point of
  and terminal point
  a vector may be represented as

This last symbol
   has special significance. It is called the standard position. The position vector has an initial point
  and a terminal point
  To change any vector into the position vector, we think about the change in the x-coordinates and the change in the y-coordinates. Thus, if the initial point of a vector
  is
  and the terminal point is
  then the position vector is found by calculating
In , we see the original vector
  and the position vector

### Finding Magnitude and Direction

To work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.

### Performing Vector Addition and Scalar Multiplication

Now that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector
  as an arrow or directed line segment from the origin to the point
  vectors can be situated anywhere in the plane. The sum of two vectors  and , or vector addition, produces a third vector + , the resultant vector.
To find  + , we first draw the vector , and from the terminal end of , we drawn the vector . In other words, we have the initial point of  meet the terminal end of . This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum  +  is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of  to the end of  in a straight path, as shown in .
Vector subtraction is similar to vector addition. To find  − , view it as  + (−). Adding − is reversing direction of  and adding it to the end of  . The new vector begins at the start of   and stops at the end point of −. See  for a visual that compares vector addition and vector subtraction using parallelograms.

### Multiplying By a Scalar

While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.

### Finding Component Form

In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the
  direction, and the vertical component is the
  direction. For example, we can see in the graph in  that the position vector
  comes from adding the vectors 1 and 2. We have 1 with initial point
  and terminal point

We also have 2 with initial point
  and terminal point

Therefore, the position vector is
Using the Pythagorean Theorem, the magnitude of 1 is 2, and the magnitude of 2 is 3. To find the magnitude of , use the formula with the position vector.
The magnitude of  is
  To find the direction, we use the tangent function

Thus, the magnitude of is
  and the direction is
  off the horizontal.

### Finding the Unit Vector in the Direction of v

In addition to finding a vector’s components, it is also useful in solving problems to find a vector in the same direction as the given vector, but of magnitude 1. We call a vector with a magnitude of 1 a unit vector. We can then preserve the direction of the original vector while simplifying calculations.
Unit vectors are defined in terms of components. The horizontal unit vector is written as
  and is directed along the positive horizontal axis. The vertical unit vector is written as
  and is directed along the positive vertical axis. See .

### Performing Operations with Vectors in Terms of i and j

So far, we have investigated the basics of vectors: magnitude and direction, vector addition and subtraction, scalar multiplication, the components of vectors, and the representation of vectors geometrically. Now that we are familiar with the general strategies used in working with vectors, we will represent vectors in rectangular coordinates in terms of  and .

### Performing Operations on Vectors in Terms of i and j

When vectors are written in terms of and we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.

### Calculating the Component Form of a Vector: Direction

We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using
  For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction.
Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with replacing

### Finding the Dot Product of Two Vectors

As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.
The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.

### Key Concepts

1. The position vector has its initial point at the origin. See .
2. If the position vector is the same for two vectors, they are equal. See .
3. Vectors are defined by their magnitude and direction. See .
4. If two vectors have the same magnitude and direction, they are equal. See .
5. Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements. See .
6. Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same. See  and .
7. Vectors are comprised of two components: the horizontal component along the positive x-axis, and the vertical component along the positive y-axis. See .
8. The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.
9. The magnitude of a vector in the rectangular coordinate system is
  See .
10. In the rectangular coordinate system, unit vectors may be represented in terms of   and   where represents the horizontal component and represents the vertical component. Then,  = a + b  is a scalar multiple of by real numbers
  See  and .
11. Adding and subtracting vectors in terms of i and j consists of adding or subtracting corresponding coefficients of i and corresponding coefficients of j. See .
12.  A vector v = a + b is written in terms of magnitude and direction as
  See .
13. The dot product of two vectors is the product of the terms plus the product of the terms. See .
14. We can use the dot product to find the angle between two vectors.  and .
15. Dot products are useful for many types of physics applications. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the two vectors and are equal, where has an initial point
  and a terminal point
  and   has an initial point
  and a terminal point
 .
For the following exercises, use the vectors  =  + 5,  = −2− 3,  and  = 4 − .
For the following exercises, use the given vectors to compute  + ,  − , and 2 − 3.
For the following exercises, find a unit vector in the same direction as the given vector.
For the following exercises, find the magnitude and direction of the vector,

### Graphical

For the following exercises, given draw 3 and

For the following exercises, use the vectors shown to sketch  + ,  − , and 2.

For the following exercises, use the vectors shown to sketch 2 + .
For the following exercises, use the vectors shown to sketch  − 3.
For the following exercises, write the vector shown in component form.

### Extensions

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

### Real-World Applications

### Chapter Review Exercises

### Non-right Triangles: Law of Sines

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.

### Non-right Triangles: Law of Cosines

### Polar Coordinates

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.
For the following exercises, convert to rectangular form and graph.

### Polar Coordinates: Graphs

For the following exercises, test each equation for symmetry.

### Polar Form of Complex Numbers

For the following exercises, find the absolute value of each complex number.
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 the product
  in polar form.
For the following exercises, find the quotient
  in polar form.
For the following exercises, find the powers of each complex number in polar form.
For the following exercises, evaluate each root.
For the following exercises, plot the complex number in the complex plane.

### Parametric Equations

For the following exercises, eliminate the parameter
  to rewrite the parametric equation as a Cartesian equation.

### Parametric Equations: Graphs

For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

### Vectors

For the following exercises, determine whether the two vectors, and are equal, where has an initial point
  and a terminal point
  and has an initial point
  and a terminal point

For the following exercises, use the vectors and to evaluate the expression.
For the following exercises, find a unit vector in the same direction as the given vector.
For the following exercises, find the magnitude and direction of the vector.
For the following exercises, calculate

### Practice Test

Given
  and
  evaluate each expression.
For the following exercises, use the vectors  =  − 3 and  = 2 + 3.


# Systems of Equations and Inequalities

## Introduction to Systems of Equations and Inequalities

At the start of the Second World War, British military and intelligence officers recognized that defeating Nazi Germany would require the Allies to know what the enemy was planning. This task was complicated by the fact that the German military transmitted all of its communications through a presumably uncrackable code created by a machine called Enigma. The Germans had been encoding their messages with this machine since the early 1930s, and were so confident in its security that they used it for everyday military communications as well as highly important strategic messages. Concerned about the increasing military threat, other European nations began working to decipher the Enigma codes. Poland was the first country to make significant advances when it trained and recruited a new group of codebreakers: math students from Poznań University. With the help of intelligence obtained by French spies, Polish mathematicians, led by Marian Rejewski, were able to decipher initial codes and later to understand the wiring of the machines; eventually they create replicas. However, the German military eventually increased the complexity of the machines by adding additional rotors, requiring a new method of decryption.
The machine attached letters on a keyboard to three, four, or five rotors (depending on the version), each with 26 starting positions that could be set prior to encoding; a decryption code (called a cipher key) essentially conveyed these settings to the message recipient, and allowed people to interpret the message using another Enigma machine. Even with the simpler three-rotor scrambler, there were 17,576 different combinations of starting positions (26 x 26 x 26); plus the machine had numerous other methods of introducing variation. Not long after the war started, the British recruited a team of brilliant codebreakers to crack the Enigma code. The codebreakers, led by Alan Turing, used what they knew about the Enigma machine to build a mechanical computer that could crack the code. And that knowledge of what the Germans were planning proved to be a key part of the ultimate Allied victory of Nazi Germany in 1945.
The Enigma is perhaps the most famous cryptographic device ever known. It stands as an example of the pivotal role cryptography has played in society. Now, technology has moved cryptanalysis to the digital world.
Many ciphers are designed using invertible matrices as the method of message transference, as finding the inverse of a matrix is generally part of the process of decoding. In addition to knowing the matrix and its inverse, the receiver must also know the key that, when used with the matrix inverse, will allow the message to be read.
In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses.


# Systems of Equations and Inequalities

## Systems of Linear Equations: Two Variables

### Learning Objectives

1. Determine whether an ordered pair is a solution of a system of equations (IA 4.1.1)
2. Solve a system of linear equations by graphing (IA 4.1.2)

### Objective: Determine whether an ordered pair is a solution of a system of equations (IA 4.1.1)

A system of linear equations is a group of two or more linear equations.  For example,

is a system of linear equations

A solution to a system of linear equations is an ordered pair x,y that is a solution to every equation in the system.

### Practice Makes Perfect

Determine whether the ordered pairs are solutions to the given system.

 at  and

### Solve a system of linear equations by graphing (IA 4.1.2)

### Practice Makes Perfect

A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible? In this section, we will consider linear equations with two variables to answer these and similar questions.

### Introduction to Systems of Equations

In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.
In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables.
The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists.
In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. A consistent system of equations has at least one solution. A consistent system is considered to be an independent system if it has a single solution, such as the example we just explored. The two lines have different slopes and intersect at one point in the plane. A consistent system is considered to be a dependent system if the equations have the same slope and the same y-intercepts. In other words, the lines coincide so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions.
Another type of system of linear equations is an inconsistent system, which is one in which the equations represent two parallel lines. The lines have the same slope and different y-intercepts. There are no points common to both lines; hence, there is no solution to the system.

###  Solving Systems of Equations by Graphing

There are multiple methods of solving systems of linear equations. For a system of linear equations in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.

### Solving Systems of Equations by Substitution

Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will consider two more methods of solving a system of linear equations that are more precise than graphing. One such method is solving a system of equations by the substitution method, in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the reason the substitution method is both valuable and practical.

### Solving Systems of Equations in Two Variables by the Addition Method

A third method of solving systems of linear equations is the addition method. In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition.

### Identifying Inconsistent Systems of Equations Containing Two Variables

Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system consists of parallel lines that have the same slope but different
  -intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as

### Expressing the Solution of a System of Dependent Equations Containing Two Variables

Recall that a dependent system of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as

###  Using Systems of Equations to Investigate Profits

Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. The skateboard manufacturer’s revenue function is the function used to calculate the amount of money that comes into the business. It can be represented by the equation
  where
  quantity and
  price. The revenue function is shown in orange in .
The cost function is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost function is shown in blue in . The
  -axis represents quantity in hundreds of units. The y-axis represents either cost or revenue in hundreds of dollars.
The point at which the two lines intersect is called the break-even point. We can see from the graph that if 700 units are produced, the cost is $3,300 and the revenue is also $3,300. In other words, the company breaks even if they produce and sell 700 units. They neither make money nor lose money.
The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss. The profit function is the revenue function minus the cost function, written as
  Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.

### Key Concepts

1. A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously.
2. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. See .
3. Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with no solution.
4. One method of solving a system of linear equations in two variables is by graphing. In this method, we graph the equations on the same set of axes. See .
5. Another method of solving a system of linear equations is by substitution. In this method, we solve for one variable in one equation and substitute the result into the second equation. See .
6. A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding opposite coefficients of corresponding variables. See .
7. It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when adding the two equations together. See , , and .
8. Either method of solving a system of equations results in a false statement for inconsistent systems because they are made up of parallel lines that never intersect. See .
9. The solution to a system of dependent equations will always be true because both equations describe the same line. See .
10. Systems of equations can be used to solve real-world problems that involve more than one variable, such as those relating to revenue, cost, and profit. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.
For the following exercises, solve each system by substitution.
For the following exercises, solve each system by addition.
For the following exercises, solve each system by any method.

### Graphical

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

### Technology

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

### Extensions

For the following exercises, solve each system in terms of
  and
  where
  are nonzero numbers. Note that
  and

### Real-World Applications

For the following exercises, solve for the desired quantity.
For the following exercises, use a system of linear equations with two variables and two equations to solve.


# Systems of Equations and Inequalities

## Systems of Linear Equations: Three Variables

### Learning Objectives

1. Determine whether an ordered triple is a solution of a system of three linear equations with three variables (IA 4.4.1)

2. Solve a system of three linear equations with three variables (IA 4.4.2)

### Objective 1: Determine whether an ordered triple is a solution of a system of three linear equations with three variables (IA 4.4.1)

A linear equation with three variables where a, b, c, and d are real numbers and a, b, and c are not all 0, is of the form  . The graph of a linear equation with three variables is a plane.

A system of linear equations with three variables is a set of linear equations with three variables. For example,

is a system of linear equations with three variables.
Solutions of a system of equations are the values of the variables that make all the equations true. A solution is represented by an ordered triple (x,y,z).

### Practice Makes Perfect

Determine whether the ordered pairs are solutions to the given system.

### Objective 2: Solve a system of three linear equations with three variables (IA 4.4.2)

When we solve a system of linear equations with three variables, we have many possible solutions.

The solutions are summarized in the table below.

### Practice Makes Perfect

Jordi received an inheritance of $12,000 that he divided into three parts and invested in three ways: in a money-market fund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual interest. Jordi invested $4,000 more in mutual funds than in municipal bonds. He earned $670 in interest the first year. How much did Jordi invest in each type of fund?
Understanding the correct approach to setting up problems such as this one makes finding a solution a matter of following a pattern. We will solve this and similar problems involving three equations and three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of visualization.

### Solving Systems of Three Equations in Three Variables

In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination, named after the prolific German mathematician Karl Friedrich Gauss. While there is no definitive order in which operations are to be performed, there are specific guidelines as to what type of moves can be made. We may number the equations to keep track of the steps we apply. The goal is to eliminate one variable at a time to achieve upper triangular form, the ideal form for a three-by-three system because it allows for straightforward back-substitution to find a solution
  which we call an ordered triple. A system in upper triangular form looks like the following:
The third equation can be solved for
  and then we back-substitute to find

 and

 To write the system in upper triangular form, we can perform the following operations:
1. Interchange the order of any two equations.
2. Multiply both sides of an equation by a nonzero constant.
3. Add a nonzero multiple of one equation to another equation.
The solution set to a three-by-three system is an ordered triple

 Graphically, the ordered triple defines the point that is the intersection of three planes in space. You can visualize such an intersection by imagining any corner in a rectangular room.  A corner is defined by three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.

### Identifying Inconsistent Systems of Equations Containing Three Variables

Just as with systems of equations in two variables, we may come across an inconsistent system of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as

 or some other contradiction.

### Expressing the Solution of a System of Dependent Equations Containing Three Variables

We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.

### Key Concepts

1. A solution set is an ordered triple

 that represents the intersection of three planes in space. See .
2. A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated.   The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. See .
3. Systems of three equations in three variables are useful for solving many different types of real-world problems. See .
4. A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction. See .
5. Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
6. A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity. See .
7. Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.
For the following exercises, solve each system by elimination.
For the following exercises, solve each system by Gaussian elimination.

### Extensions

For the following exercises, solve the system for
  and

### Real-World Applications