The Unit Circle: Sine and Cosine Functions

# Unit Circle

### Learning Objectives

In this section you will:

- Find function values for the sine and cosine ofand
- Identify the domain and range of sine and cosine functions.
- Find reference angles.
- Use reference angles to evaluate trigonometric functions.

Looking for a thrill? Then consider a ride on the Singapore Flyer, the world’s tallest Ferris wheel. Located in Singapore, the Ferris wheel soars to a height of 541 feet—a little more than a tenth of a mile! Described as an observation wheel, riders enjoy spectacular views as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs.

### Finding Trigonometric Functions Using the Unit Circle

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

The *x-* and *y-*axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.

For any anglewe can label the intersection of the terminal side and the unit circle as by its coordinates,The coordinatesandwill be the outputs of the trigonometric functionsandrespectively. This meansand

### Unit Circle

A unit circle has a center atand radiusIn a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle

Letbe the endpoint on the unit circle of an arc of arc lengthThecoordinates of this point can be described as functions of the angle.

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

The sine function relates a real numberto the *y*-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angleequals the *y*-value of the endpoint on the unit circle of an arc of lengthIn (Figure), the sine is equal toLike 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 angleequals the *x*-value of the endpoint on the unit circle of an arc of lengthIn (Figure), 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 asandis the same asLikewise,is a commonly used shorthand notation forBe 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.

### Sine and Cosine Functions

Ifis a real number and a pointon the unit circle corresponds to a central anglethen

### How To

**Given a point Pon the unit circle corresponding to an angle offind the sine and cosine.**

- The sine ofis equal to the
*y*-coordinate of point - The cosine ofis equal to the
*x*-coordinate of point

### Finding Function Values for Sine and Cosine

Pointis a point on the unit circle corresponding to an angle ofas shown in (Figure). Findand

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We know thatis the *x*-coordinate of the corresponding point on the unit circle andis the *y*-coordinate of the corresponding point on the unit circle. So:

### Try It

A certain anglecorresponds to a point on the unit circle atas shown in (Figure). Findand

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#### 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 ofand

### Calculating Sines and Cosines along an Axis

Findand

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Movingcounterclockwise around the unit circle from the positive *x*-axis brings us to the top of the circle, where thecoordinates areas shown in (Figure).

We can then use our definitions of cosine and sine.

The cosine ofis 0; the sine ofis 1.[/hidden-answer]

### Try It

Find cosine and sine of the angle

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#### 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 isBecauseandwe can substitute forandto getThis equation,is known as the Pythagorean Identity. See (Figure).

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.

### Pythagorean Identity

The Pythagorean Identity states that, for any real number

### How To

**Given the sine of some angleand its quadrant location, find the cosine of
**

- Substitute the known value ofinto the Pythagorean Identity.
- Solve for
- Choose the solution with the appropriate sign for the
*x*-values in the quadrant whereis located.

### Finding a Cosine from a Sine or a Sine from a Cosine

Ifandis in the second quadrant, find

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If we drop a vertical line from the point on the unit circle corresponding towe create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. See (Figure).

Substituting the known value for sine into the Pythagorean Identity,

Because the angle is in the second quadrant, we know the *x-*value is a negative real number, so the cosine is also negative.

### Try It

Ifandis in the fourth quadrant, find

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### Finding Sines and Cosines of Special Angles

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

#### Finding Sines and Cosines ofAngles

First, we will look at angles oforas shown in (Figure). Atriangle 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.

Atwhich is 45 degrees, the radius of the unit circle bisects the first quadrantal angle. This means the radius lies along the lineA unit circle has a radius equal to 1 so the right triangle formed below the linehas sidesandand radius = 1. See (Figure).

From the Pythagorean Theorem we get

We can then substitute

Next we combine like terms.

And solving forwe get

In quadrant I,

Ator 45 degrees,

If we then rationalize the denominators, we get

Therefore, thecoordinates of a point on a circle of radiusat an angle ofare

#### Finding Sines and Cosines ofandAngles

Next, we will find the cosine and sine at an angle oforFirst, we will draw a triangle inside a circle with one side at an angle ofand another at an angle ofas shown in (Figure). If the resulting two right triangles are combined into one large triangle, notice that all three angles of this larger triangle will beas shown in (Figure).

Because all the angles are equal, the sides are also equal. The vertical line has lengthand since the sides are all equal, we can also conclude thatorSince

And sincein our unit circle,

Using the Pythagorean Identity, we can find the cosine value.

Thecoordinates for the point on a circle of radiusat an angle ofareAtthe radius of the unit circle, 1, serves as the hypotenuse of a 30-60-90 degree right triangle,as shown in (Figure). Anglehas measureAt pointwe draw an anglewith measure ofWe know the angles in a triangle sum toso the measure of angleis alsoNow we have an equilateral triangle. Because each side of the equilateral triangleis 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 angleis 30°. Angleis double angleso its measure is 60°.is the perpendicular bisector ofso it cutsin half. This means thatisthe radius, orNotice thatis the *x*-coordinate of pointwhich is at the intersection of the 60° angle and the unit circle. This gives us a trianglewith hypotenuse of 1 and sideof length

From the Pythagorean Theorem, we get

Substitutingwe get

Solving forwe get

Sincehas the terminal side in quadrant I where the *y-*coordinate is positive, we choosethe positive value.

At(60°), thecoordinates for the point on a circle of radiusat an angle ofareso 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. (Figure) summarizes these values.

Angle |
or | or | or | or | |

Cosine |
1 | 0 | |||

Sine |
0 | 1 |

(Figure) 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 evaluateon 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.

### How To

**Given an angle in radians, use a graphing calculator to find the cosine.
**

- If the calculator has degree mode and radian mode, set it to radian mode.
- Press the COS key.
- Enter the radian value of the angle and press the close-parentheses key “)”.
- Press ENTER.

### Using a Graphing Calculator to Find Sine and Cosine

Evaluateusing a graphing calculator or computer.

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Enter the following keystrokes:

#### Analysis

We can find the cosine or sine of an angle in degrees directly on a calculator with degree mode. For calculators or software that use only radian mode, we can find the sign offor example, by including the conversion factor to radians as part of the input:

### Try It

Evaluate

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approximately 0.866025403

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### 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 thanand angles larger thancan still be graphed on the unit circle and have real values ofthere 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 (Figure). The bounds of the *x*-coordinate areThe bounds of the *y*-coordinate are alsoTherefore, 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 (Figure), anglehas the same sine value as anglethe cosine values are opposites. Anglehas the same cosine value as anglethe sine values are opposites.

Recall that an angle’s reference angle is the acute angle,formed by the terminal side of the angleand the horizontal axis. A reference angle is always an angle betweenandorandradians. As we can see from (Figure), for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.

### How To

**Given an angle betweenandfind its reference angle.**

- An angle in the first quadrant is its own reference angle.
- For an angle in the second or third quadrant, the reference angle isor
- For an angle in the fourth quadrant, the reference angle isor
- If an angle is less thanor greater thanadd or subtractas many times as needed to find an equivalent angle betweenand

### Finding a Reference Angle

Find the reference angle ofas shown in (Figure).

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Becauseis in the third quadrant, the reference angle is

### Try It

Find the reference angle of

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### 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 findcoordinates 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 Cosine and Sine

Angles have cosines and sines with the same absolute value as their reference angles. The sign (positive or negative) can be determined from the quadrant of the angle.

### How To

**Given an angle in standard position, find the reference angle, and the cosine and sine of the original angle.
**

- Measure the angle between the terminal side of the given angle and the horizontal axis. That is the reference angle.
- Determine the values of the cosine and sine of the reference angle.
- Give the cosine the same sign as the
*x*-values in the quadrant of the original angle. - Give the sine the same sign as the
*y*-values in the quadrant of the original angle.

### Using Reference Angles to Find Sine and Cosine

- Using a reference angle, find the exact value ofand
- Using the reference angle, findand

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- is located in the second quadrant. The angle it makes with the
*x*-axis isso the reference angle isThis tells us thathas the same sine and cosine values asexcept for the sign.

Sinceis in the second quadrant, the

*x*-coordinate of the point on the circle is negative, so the cosine value is negative. The*y*-coordinate is positive, so the sine value is positive. - is in the third quadrant. Its reference angle isThe cosine and sine ofare bothIn the third quadrant, bothandare negative, so:
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### Try It

- Use the reference angle ofto find and
- Use the reference angle ofto findand

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#### 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 (Figure). Take time to learn the coordinates of all of the major angles in the first quadrant.

### Key Equations

Cosine | |

Sine | |

Pythagorean Identity |

### Key Concepts

- 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.
- Using the unit circle, the sine of an angleequals the
*y*-value of the endpoint on the unit circle of an arc of lengthwhereas the cosine of an angleequals the*x*-value of the endpoint. See (Figure). - The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. See (Figure).
- 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 (Figure).
- Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. See (Figure).
- The domain of the sine and cosine functions is all real numbers.
- The range of both the sine and cosine functions is
- The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.
- The signs of the sine and cosine are determined from the
*x*– and*y*-values in the quadrant of the original angle. - An angle’s reference angle is the size angle,formed by the terminal side of the angleand the horizontal axis. See (Figure).
- Reference angles can be used to find the sine and cosine of the original angle. See (Figure).
- Reference angles can also be used to find the coordinates of a point on a circle. See (Figure).

### Section Exercises

#### Verbal

Describe the unit circle.

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The unit circle is a circle of radius 1 centered at the origin.

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What do the *x-* and *y-*coordinates of the points on the unit circle represent?

Discuss the difference between a coterminal angle and a reference angle.

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Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle,formed by the terminal side of the angleand the horizontal axis.

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Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.

Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.

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The sine values are equal.

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#### 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 bylies.

and

and

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I

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and

and

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IV

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For the following exercises, find the exact value of each trigonometric function.

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0

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-1

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#### Numeric

For the following exercises, state the reference angle for the given angle.

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

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Quadrant IV,

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Quadrant II,

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Quadrant II,

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Quadrant II,

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Quadrant III,

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Quadrant II,

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Quadrant II,

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Quadrant IV,

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For the following exercises, find the requested value.

Ifandis in the fourth quadrant, find

Ifandis in the first quadrant, find

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Ifandis in the second quadrant, find

Ifandis in the third quadrant, find

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Find the coordinates of the point on a circle with radius 15 corresponding to an angle of

Find the coordinates of the point on a circle with radius 20 corresponding to an angle of

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Find the coordinates of the point on a circle with radius 8 corresponding to an angle of

Find the coordinates of the point on a circle with radius 16 corresponding to an angle of

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State the domain of the sine and cosine functions.

State the range of the sine and cosine functions.

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#### Graphical

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

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#### Technology

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

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−0.1736

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0.9511

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−0.7071

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−0.1392

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−0.7660

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#### Extensions

For the following exercises, evaluate.

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0

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

What are the coordinates of the child after 45 seconds?

What are the coordinates of the child after 90 seconds?

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What are the coordinates of the child after 125 seconds?

When will the child have coordinatesif the ride lasts 6 minutes? (There are multiple answers.)

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37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

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When will the child have coordinatesif the ride lasts 6 minutes?

### Glossary

- cosine function
- the
*x*-value of the point on a unit circle corresponding to a given angle

- Pythagorean Identity
- a corollary of the Pythagorean Theorem stating that the square of the cosine of a given angle plus the square of the sine of that angle equals 1

- sine function
- the
*y*-value of the point on a unit circle corresponding to a given angle