# The Unit Circle: Sine and Cosine Functions

## Unit Circle

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

### Finding Trigonometric Functions Using the Unit Circle

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


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



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

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


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

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

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



### The Pythagorean Identity

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

### Finding Sines and Cosines of Special Angles

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

### Finding Sines and Cosines of   Angles

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


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


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



### Finding Sines and Cosines of   and   Angles 

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

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


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

### Using a Calculator to Find Sine and Cosine

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

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

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



### Finding Reference Angles

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

### Using Reference Angles 

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

### Using Reference Angles to Evaluate Trigonometric Functions

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

### Using Reference Angles to Find Coordinates

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

### Key Equations


### Key Concepts

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


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

### Section Exercises


### Verbal


### Algebraic

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

### Numeric

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

### Graphical

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

### Technology

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

### Extensions

For the following exercises, evaluate.

### Real-World Applications

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