### Learning Outcomes

- Identify reference angles for angles measured in both radians and degrees
- Evaluate trigonometric functions using the unit circle

In the Derivatives of Trigonometric Functions section, you will sometimes be asked to evaluate the derivatives of trigonometric functions at specific values. Here we will review how to evaluate trigonometric functions at various angle measures.

## Find Reference Angles

(also in Module 1, Skills Review for Trigonometric Functions)

You learned that it is easiest to evaluate trigonometric functions when an angle is in the first quadrant. When the original angle is given in quadrant two, three, or four, a reference angle should be found.

An angle’s reference angle is the acute angle, [latex]t[/latex], formed by the terminal side of the angle [latex]t[/latex] and the horizontal axis. A reference angle is always an angle between [latex]0[/latex] and [latex]90^\circ [/latex], or [latex]0[/latex] and [latex]\frac{\pi }{2}[/latex] radians. As we can see in the figure below, for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.

A visual of the corresponding reference angles for each of the quadrants.

How To: Given an angle between [latex]0[/latex] and [latex]2\pi [/latex], find 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 is [latex]|\pi -t|[/latex] or [latex]|180^\circ \mathrm{-t}|[/latex].
- For an angle in the fourth quadrant, the reference angle is [latex]2\pi -t[/latex] or [latex]360^\circ \mathrm{-t}[/latex].
- If an angle is less than [latex]0[/latex] or greater than [latex]2\pi [/latex], add or subtract [latex]2\pi [/latex] as many times as needed to find an equivalent angle between [latex]0[/latex] and [latex]2\pi [/latex].

### Example: Finding a Reference Angle

Find the reference angle of [latex]225^\circ [/latex] as shown in below.

Because [latex]225^\circ [/latex] is in the third quadrant, the reference angle is

[latex]|\left(180^\circ -225^\circ \right)|=|-45^\circ |=45^\circ [/latex]

### Try It

Find the reference angle of [latex]\frac{5\pi }{3}[/latex].

[latex]\frac{\pi }{3}[/latex]

We can evaluate trigonometric functions of angles outside the first quadrant using reference angles. The quadrant of the original angle determines whether the answer is positive or negative. 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,” all of the six trigonometric functions are positive. In quadrant II, “Smart,” only sine and its reciprocal function, cosecant, are positive. In quadrant III, “Trig,” only tangent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “Class,” only cosine and its reciprocal function, secant, are positive.

An illustration of which trigonometric functions are positive in each of the quadrants.

## Evaluate Trigonometric Functions Using the Unit Circle

(also in Module 1, Skills Review for Trigonometric Functions)

The unit circle tells us the value of cosine and sine at any of the given angle measures seen below. The first coordinate in each ordered pair is the value of cosine at the given angle measure, while the second coordinate in each ordered pair is the value of sine at the given angle measure. You learned in Section 1.3 that all trigonometric functions can be written in terms of sine and cosine. Thus, if you can evaluate sine and cosine at various angle values, you can also evaluate the other trigonometric functions at various angle values. Take time to learn the [latex]\left(x,y\right)[/latex] coordinates of all of the major angles in the first quadrant of the unit circle.

Remember, every angle in quadrant two, three, or four has a reference angle that lies in quadrant one. The quadrant of the original angle only affects the sign (positive or negative) of a trigonometric function’s value at a given angle.

How To: Given the angle of a point on The Unit circle, find the Value of Cosine (Or Sine) using quadrant one.

- Find the reference angle using the appropriate reference angle formula from the first portion of this review section.
- Find the value of cosine (or sine) at the reference angle by looking at quadrant one of the unit circle.
- Determine the appropriate sign of your found value for cosine (or sine) based on the quadrant of the original angle.

Example: Using the Unit Circle to Find the Value of cosine

Use quadrant one of the unit circle to find the value of cosine at an angle of [latex]\frac{7\pi }{6}[/latex].

We know that the angle [latex]\frac{7\pi }{6}[/latex] is in the third quadrant.

First, let’s find the reference angle. The reference angle is:

[latex]\frac{7\pi }{6}-\pi =\frac{\pi }{6}[/latex]

Next, we find the value of cosine at the reference angle which is represented by the first coordinate of the ordered pair at [latex]\frac{\pi }{6}[/latex].

[latex]\cos \left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}[/latex]

Because our original angle is in the third quadrant, where cosine is always negative, we have:

[latex]\cos \left(\frac{7\pi }{6}\right)=-\frac{\sqrt{3}}{2}[/latex]

### Try It

Use quadrant one of the unit circle to find the value of sine at an angle of [latex]\frac{5\pi }{3}[/latex].

[latex]-\frac{\sqrt{3}}{2}[/latex]