# 4 Unit Circle

### Learning Objectives

In this section you will:

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

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### 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) 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

### Unit Circle

A unit circle has a center at and radius In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle

Let be the endpoint on the unit circle of an arc of arc length The coordinates 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 number[ latex]\,t\,[/latex] 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 (Figure), 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 (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 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.

### Sine and Cosine Functions

If is a real number and a point on the unit circle corresponds to a central angle then

### How To

**Given a point P on the unit circle corresponding to an angle of find the sine and cosine.**

- The sine of i s equal to the
*y*-coordinate of point - The cosine of is equal to the
*x*-coordinate of point

### Finding Function Values for Sine and Cosine

A point on the unit circle corresponds to an angle of as shown in (Figure). Find and

## Show Solution

We know that is the *x*-coordinate of the corresponding point on the unit circle and is the *y*-coordinate of the corresponding point on the unit circle. So:

### Try It

A certain angle corresponds to a point on the unit circle at as shown in (Figure). Find and

## Show Solution

#### 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

### Calculating Sines and Cosines along an Axis

Find and

## Show Solution

Moving counterclockwise around the unit circle from the positive *x*-axis brings us to the top of the circle, where the coordinates are as shown in (Figure).

We can then use our definitions of cosine and sine.

The cosine of is 0; the sine of is 1.

### Try It

Find cosine and sine of the angle

## Show Solution

#### 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 (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 angle and its quadrant location, find the cosine of
**

- Substitute the known value of into the Pythagorean Identity.
- Solve for
- Choose the solution with the appropriate sign for the
*x*-values in the quadrant where is located.

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

If and is in the second quadrant, find

## Show Solution

If we drop a vertical line from the point on the unit circle corresponding to we 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

If and is in the fourth quadrant, find