Lesson Objectives
• Learn how to find reference angles
• Learn how to find trigonometric function values for non-acute angles

## How to Find Reference Angles

In this lesson, we will learn how to find trigonometric function values of non-acute angles. We will begin by discussing the concept of reference angles. Recall that quadrantal angles are angles in standard position with measures that are multiples of 90°: (90°, 180°, 270°,...). Every nonquadrantal angle in standard position will have a positive acute angle known as its reference angle. The reference angle for the angle θ is written as θ' (read as theta prime). θ' is the positive acute angle that is made by the terminal side of our angle θ and the x-axis. To see this more clearly, let's look at three diagrams, one for quadrants II, III, and IV. Note that in quadrant I, θ and θ' are the same.

### Reference Angle θ' for θ, where 0° < θ < 360°:

Q Iθ' = θ
Q IIθ' = 180° - θ
Q IIIθ' = θ - 180°
Q IVθ' = 360° - θ

θ' = 180° - θ

θ' = θ - 180°

### Reference Angle in Quadrant IV

θ' = 360° - θ Let's look at an example.
Example #1: Find the reference angle for each angle.
330°
Since 330° is between 0° and 360° and lies in quadrant IV, we can find the reference angle by subtracting 360° - 330°.
360° - 330° = 30° When our angle θ is negative or has a measure that is greater than 360°, its reference angle is found by finding its coterminal angle that is between 0° and 360°. Let's look at a few examples.
Example #2: Find the reference angle for each angle.
-250°
Since -250° is negative, we first need to find a coterminal angle that is between 0° and 360°. Let's add 360° to -250°.
-250° + 360° = 110°
Now, we will find the reference angle for 110°. Since this angle lies in quadrant II we will subtract 180° - 110°.
180° - 110° = 70° Example #3: Find the reference angle for each angle.
560°
Since 560° is greater than 360°, we first need to find a coterminal angle that is between 0° and 360°. Let's subtract 360° from 560°.
560° - 360° = 200°
Now, we will find the reference angle for 200°. Since this angle lies in quadrant III, we will subtract 200° - 180°.
200° - 180° = 20°

## Finding Trigonometric Function Values for Any Nonquadrantal Angle θ

1. Find the reference angle θ'
2. Find the trigonometric function values for reference angle θ'
3. Use the sign rules to determine the correct signs for each function

### Trigonometric Function Values of Special Angles

Certain angles appear very frequently: 30°, 45°, and 60°. The function values of these special angles can be summarized using the following table:
θ sin θ cos θ tan θ
30°$\frac{1}{2}$$\frac{\sqrt{3}}{2}$$\frac{\sqrt{3}}{3}$
45°$\frac{\sqrt{2}}{2}$$\frac{\sqrt{2}}{2}$$1$
60°$\frac{\sqrt{3}}{2}$$\frac{1}{2}$$\sqrt{3}$
The other three function values can be found using the reciprocal identities:
θ cot θ sec θ csc θ
30°$\sqrt{3}$$\frac{2\sqrt{3}}{3}$$2$
45°$1$$\sqrt{2}$$\sqrt{2}$
60°$\frac{\sqrt{3}}{3}$$2$$\frac{2\sqrt{3}}{3}$
Let's look at a few examples.
Example 4: Use the table above to find the exact value of each expression.
cos(-600°)
First, we want to find our reference angle. Since we have a -600° angle, we first find a coterminal angle between 0° and 360°.
-600° + 2 • 360° = -600° + 720° = 120°
Since a 120° angle lies in quadrant II, we want to subtract 180° - 120°.
180° - 120° = 60°
Now, we will find the value of cos(60°). To do this, we can reference our table above. $$\text{cos}(60°)=\frac{1}{2}$$ Lastly, let's use our sign rules to determine the correct sign. Since our angle -600° or its coterminal angle 120° lies in quadrant II, we know that cos θ is negative. Let's obtain our final answer by changing the sign: $$\text{cos}(-600°)=-\text{cos}(60°)=-\frac{1}{2}$$ Example 5: Use the table above to find the exact value of each expression.
sec(495°)
First, we want to find our reference angle. Since we have a 495° angle, we first find a coterminal angle between 0° and 360°.
495° - 360° = 135°
Since a 135° angle lies in quadrant II, we want to subtract 180° - 135°.
180° - 135° = 45°
Now, we will find the value of sec(45°). To do this, we can reference our table above. $$\text{sec}(45°)=\sqrt{2}$$ Lastly, let's use our sign rules to determine the correct sign. Since our angle 495° or its coterminal angle 135° lies in quadrant II, we know that sec θ is negative. Let's obtain our final answer by changing the sign: $$\text{sec}(495°)=-\text{sec}(45°)=-\sqrt{2}$$

#### Skills Check:

Example #1

Find the reference angle.

-195°

A
85°
B
15°
C
25°
D
20°
E
-165°

Example #2

Find the exact value.

cot(-300°)

A
$$\text{cot}(-300°)=-1$$
B
$$\text{cot}(-300°)=1$$
C
$$\text{cot}(-300°)=2$$
D
$$\text{cot}(-300°)=\frac{\sqrt{3}}{3}$$
E
$$\text{cot}(-300°)=\frac{\sqrt{3}}{2}$$

Example #3

Find the exact value.

sec 120°

A
$$\text{sec}120°=-1$$
B
$$\text{sec}120°=\frac{\sqrt{2}}{2}$$
C
$$\text{sec}120°=-2$$
D
$$\text{sec}120°=-\frac{2\sqrt{3}}{3}$$
E
$$\text{sec}120°=-2$$