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.

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°

The other three function values can be found using the reciprocal identities:

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: $$\text{sec}(495°)=-\text{sec}(45°)=-\sqrt{2}$$

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

Quadrant | Reference Angle |
---|---|

Q I | θ' = θ |

Q II | θ' = 180° - θ |

Q III | θ' = θ - 180° |

Q IV | θ' = 360° - θ |

### Reference Angle in Quadrant II

θ' = 180° - θ### Reference Angle in Quadrant III

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

- Find the reference angle θ'
- Find the trigonometric function values for reference angle θ'
- 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° | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |

θ | cot θ | sec θ | csc θ |
---|---|---|---|

30° | $\sqrt{3}$ | $\frac{2\sqrt{3}}{3}$ | $2$ |

45° | $1$ | $\sqrt{2}$ | $\sqrt{2}$ |

60° | $\frac{\sqrt{3}}{3}$ | $2$ | $\frac{2\sqrt{3}}{3}$ |

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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