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°:

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

Reference Angle in Quadrant II

θ' = 180° - θ reference angle theta prime in quadrant II

Reference Angle in Quadrant III

θ' = θ - 180° reference angle theta prime in quadrant III

Reference Angle in Quadrant IV

θ' = 360° - θ reference angle theta prime in quadrant IV 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° reference angle for a 330 degree angle 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° reference angle for a -250 degree angle 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° reference angle for a 200 degree angle

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°

Please choose the best answer.

A
85°
B
15°
C
25°
D
20°
E
-165°

Example #2

Find the exact value.

cot(-300°)

Please choose the best answer.

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°

Please choose the best answer.

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
$$\text{sec}120°=-2$$
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