# Trigonometric Functions

## Unit Circle: Sine and Cosine Functions

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

###  Finding Function Values for the Sine and Cosine

To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in . The angle (in radians) that
    intercepts forms an arc of length
    Using the formula
    and knowing that
    we see that for a unit circle,

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

### Defining Sine and Cosine Functions

Now that we have our unit circle labeled, we can learn how the
 coordinates relate to the arc length and angle. The sine function relates a real number
 to the y-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle
 equals the y-value of the endpoint on the unit circle of an arc of length

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

Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses:
  is the same as
 and

 is the same as
 Likewise,
 is a commonly used shorthand notation for
 Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer.

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

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

### The Pythagorean Identity

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

### Finding Sines and Cosines of Special Angles

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

### Finding Sines and Cosines of 45° Angles

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

At
 or 45 degrees,

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

### Finding Sines and Cosines of 30° and 60° Angles

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

    in our unit circle,
Using the Pythagorean Identity, we can find the cosine value.
The
 coordinates for the point on a circle of radius
 at an angle of
 are

    At

(60°), the radius of the unit circle, 1, serves as the hypotenuse of a 30-60-90 degree right triangle,
 as shown in . Angle
 has measure
 At point
 we draw an angle
 with measure of
 We know the angles in a triangle sum to
 so the measure of angle
 is also
 Now we have an equilateral triangle. Because each side of the equilateral triangle
 is the same length, and we know one side is the radius of the unit circle, all sides must be of length 1.
The measure of angle
 is 30°. So, if double, angle
 is 60°.
 is the perpendicular bisector of
 so it cuts
 in half.  This means that
 is
 the radius, or
 Notice that
 is the x-coordinate of point
 which is at the intersection of the 60° angle and the unit circle. This gives us a triangle
 with hypotenuse of 1 and side
 of length

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

### Using a Calculator to Find Sine and Cosine

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

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

Now that we can find the sine and cosine of an angle, we need to discuss their domains and ranges. What are the domains of the sine and cosine functions? That is, what are the smallest and largest numbers that can be inputs of the functions? Because angles smaller than 0 and angles larger than
    can still be graphed on the unit circle and have real values of
  and
  there is no lower or upper limit to the angles that can be inputs to the sine and cosine functions. The input to the sine and cosine functions is the rotation from the positive x-axis, and that may be any real number.
What are the ranges of the sine and cosine functions? What are the least and greatest possible values for their output? We can see the answers by examining the unit circle, as shown in . The bounds of the x-coordinate are
    The bounds of the y-coordinate are also
    Therefore, the range of both the sine and cosine functions is

### Finding Reference Angles

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

### Using Reference Angles

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

### Using Reference Angles to Evaluate Trigonometric Functions

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

### Using Reference Angles to Find Coordinates

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

### Key Equations

### Key Concepts

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

8. The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.
9. The signs of the sine and cosine are determined from the x- and y-values in the quadrant of the original angle.
10. An angle’s reference angle is the size angle,

     formed by the terminal side of the angle
   and the horizontal axis. See .
11. Reference angles can be used to find the sine and cosine of the original angle. See .
12. Reference angles can also be used to find the coordinates of a point on a circle. See .

### Section Exercises

### Verbal

### Algebraic

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

### Numeric

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

### Graphical

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

### Technology

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

### Extensions

For the following exercises, evaluate.

### Real-World Applications

For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point

  that is, on the due north position.  Assume the carousel revolves counter clockwise.


# Trigonometric Functions

## The Other Trigonometric Functions

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

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

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

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

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

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

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

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

### Using Even and Odd Trigonometric Functions

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

   and so on. So
     is an even function, a function such that two inputs that are opposites have the same output. That means

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

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

###  Recognizing and Using Fundamental Identities

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

### Alternate Forms of the Pythagorean Identity

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

The other form is obtained by dividing both sides by

As we discussed in the chapter opening, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or

    will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.
Other functions can also be periodic. For example, the lengths of months repeat every four years. If
     represents the length time, measured in years, and
     represents the number of days in February, then

    This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.

### Evaluating Trigonometric Functions with a Calculator

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

### Key Equations

### Key Concepts

1. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle.
2. The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
3. The six trigonometric functions can be found from a point on the unit circle. See .
4. Trigonometric functions can also be found from an angle. See  .
5. Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See .
6. A function is said to be even if
   and odd if

7. Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
8. Even and odd properties can be used to evaluate trigonometric functions. See .
9. The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
10. Identities can be used to evaluate trigonometric functions. See  and .
11. Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See .
12. The trigonometric functions repeat at regular intervals.
13. The period
   of a repeating function
   is the smallest interval such that
   for any value of

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

### Section Exercises

### Verbal

### Algebraic

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

### Graphical

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

### Technology

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

### Extensions

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

### Real-World Applications


# Trigonometric Functions

## Right Triangle Trigonometry

We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle:
In this section, we will see another way to define trigonometric functions using properties of right triangles.

### Using Right Triangles to Evaluate Trigonometric Functions

In earlier sections, we used a unit circle to define the trigonometric functions. In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of
  is its value at
  radians. First, we need to create our right triangle.  shows a point on a unit circle of radius 1. If we drop a vertical line segment from the point
  to the x-axis, we have a right triangle whose vertical side has length
  and whose horizontal side has length
  We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.
We know
Likewise, we know
These ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using
  coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of
  we will call the side between the given angle and the right angle the adjacent side to angle
  (Adjacent means “next to.”) Instead of
  we will call the side most distant from the given angle the opposite side from angle
  And instead of
  we will call the side of a right triangle opposite the right angle the hypotenuse. These sides are labeled in .

### Understanding Right Triangle Relationships

Given a right triangle with an acute angle of

A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.”

### Relating Angles and Their Functions

When working with right triangles, the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in . The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.
We will be asked to find all six trigonometric functions for a given angle in a triangle. Our strategy is to find the sine, cosine, and tangent of the angles first. Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.

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

We have already discussed the trigonometric functions as they relate to the special angles on the unit circle. Now, we can use those relationships to evaluate triangles that contain those special angles. We do this because when we evaluate the special angles in trigonometric functions, they have relatively friendly values, values that contain either no or just one square root in the ratio. Therefore, these are the angles often used in math and science problems. We will use multiples of

  and

however, remember that when dealing with right triangles, we are limited to angles between

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

### Using Equal Cofunction of Complements

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

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

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

  then
  as well.

### Using Trigonometric Functions

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

### Using Right Triangle Trigonometry to Solve Applied Problems

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

### Key Equations

### Key Concepts

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

### Section Exercises

### Verbal

### Algebraic

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

### Graphical

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

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

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

### Technology

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

### Extensions

### Real-World Applications

### Review Exercises

### Angles

For the following exercises, convert the angle measures to degrees.
For the following exercises, convert the angle measures to radians.
For the following exercises, find the angle between 0° and 360° that is coterminal with the given angle.
For the following exercises, find the angle between 0 and
  in radians that is coterminal with the given angle.
For the following exercises, draw the angle provided in standard position on the Cartesian plane.

### Unit Circle: Sine and Cosine Functions

### The Other Trigonometric Functions

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

### Right Triangle Trigonometry

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

### Practice Test


# Periodic Functions

## Introduction to Periodic Functions

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


# Periodic Functions

## Graphs of the Sine and Cosine Functions

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

### Graphing Sine and Cosine Functions

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

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

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

### Investigating Sinusoidal Functions

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

### Determining the Period of Sinusoidal Functions

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

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

### Determining Amplitude

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

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

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

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

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

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

### Using Transformations of Sine and Cosine Functions

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

### Key Equations

### Key Concepts

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

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

### Section Exercises

### Verbal

### Graphical

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

### Algebraic

For the following exercises, let

For the following exercises, let

### Technology

### Real-World Applications


# Periodic Functions

## Graphs of the Other Trigonometric Functions

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

### Analyzing the Graph of y = tan x

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

### Graphing Variations of y = tan x

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

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

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

### Graphing One Period of a Shifted Tangent Function

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

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

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

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

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

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

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

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

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

### Analyzing the Graph of y = cot x

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

### Graphing Variations of y = cot x

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

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

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

### Key Equations

### Key Concepts

1. The tangent function has period

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

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

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

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

### Section Exercises

### Verbal

### Algebraic

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

### Graphical

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

### Technology

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

### Real-World Applications


# Periodic Functions

## Inverse Trigonometric Functions

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

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

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

2. Since
         then

3. Since
         then

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

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

 shows the graph of the tangent function limited to

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

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

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

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

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

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

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

### Using a Calculator to Evaluate Inverse Trigonometric Functions

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

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

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

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

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

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

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

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

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

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

### Key Concepts

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

### Section Exercises

### Verbal

### Algebraic

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

    with the help of a reference triangle.

### Extensions

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

### Graphical

### Real-World Applications

### Chapter Review Exercises

### Graphs of the Sine and Cosine Functions

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

### Graphs of the Other Trigonometric Functions

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

### Inverse Trigonometric Functions

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

### Chapter Practice Test

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

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

Evaluate the following expressions.

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


# Trigonometric Identities and Equations

## Introduction to Trigonometric Identities and Equations

Math is everywhere, even in places we might not immediately recognize. For example, mathematical relationships describe the transmission of images, light, and sound. The sinusoidal graph in the figure above models music playing on a phone, radio, or computer. Such graphs are described using trigonometric equations and functions. In this chapter, we discuss how to manipulate trigonometric equations algebraically by applying various formulas and trigonometric identities. We will also investigate some of the ways that trigonometric equations are used to model real-life phenomena.


# Trigonometric Identities and Equations

## Solving Trigonometric Equations with Identities

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

### Verifying the Fundamental Trigonometric Identities

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

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

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

### Using Algebra to Simplify Trigonometric Expressions

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

### Key Equations

### Key Concepts

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

### Section Exercises

### Verbal

### Algebraic

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

### Extensions

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


# Trigonometric Identities and Equations

## Sum and Difference Identities

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

### Using the Sum and Difference Formulas for Cosine

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

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

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

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

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

### Using the Sum and Difference Formulas for Sine

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

### Using the Sum and Difference Formulas for Tangent

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

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

### Using Sum and Difference Formulas for Cofunctions

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

Notice also that
       opposite over hypotenuse. Thus, when two angles are complementary, we can say that the sine of
       equals the cofunction of the complement of
       Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.
From these relationships, the cofunction identities are formed.
Notice that the formulas in the table may also be justified algebraically using the sum and difference formulas. For example, using
we can write

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

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

### Key Equations

### Key Concepts

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

### Section Exercises

### Verbal

### Algebraic

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

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

### Graphical

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

        )

### Technology

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

### Extensions

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


# Trigonometric Identities and Equations

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

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

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

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

### Using Double-Angle Formulas to Verify Identities

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

### Use Reduction Formulas to Simplify an Expression

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

Next, we use the formula
       Solve for

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

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

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

### Key Equations

### Key Concepts

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

### Section Exercises

### Verbal

### Algebraic

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

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

      when

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

### Technology

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

### Extensions

For the following exercises, prove the identities.


# Trigonometric Identities and Equations

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

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

### Expressing Products as Sums

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

### Expressing Products as Sums for Cosine

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

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

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

### Expressing Products of Sines in Terms of Cosine

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

### Expressing Sums as Products

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

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

### Key Equations

### Key Concepts

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

### Section Exercises

### Verbal

### Algebraic

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

### Numeric

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

### Technology

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

### Extensions

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