# 1.19: Trigonometric Functions of Negative Angles

**At Grade**Created by: CK-12

**Practice**Trigonometric Functions of Negative Angles

While practicing for the track team, you regularly stop to consider the values of trig functions for the angle you've covered as you run around the circular track at your school. Today, however, is different. To keep things more interesting, your coach has decided to have you and your teammates run the opposite of the usual direction on the track. From your studies at school, you know that this is the equivalent of a "negative angle".

You have run

### Trigonometric Functions of Negative Angles

Recall that graphing a negative angle means rotating clockwise. The graph below shows

Notice that this angle is coterminal with

In general, if a negative angle has a reference angle of

#### Find the value of the expression:

#### Find the value of the expression:

The angle

#### Find the value of the expression:

The angle

We can also use our knowledge of reference angles and ordered pairs to find the values of trig functions of angles with measure greater than 360 degrees.

### Examples

#### Example 1

Earlier, you were asked if it is still possible to find the values of trig functions for the new type of angles.

What you want to find is the value of the expression:

#### Example 2

Find the value of the expression:

The angle

#### Example 3

Find the value of the expression:

The angle

#### Example 4

Find the value of the expression:

The angle

### Review

Calculate each value.

sin−120∘ cos−120∘ tan−120∘ csc−120∘ - \begin{align*}\sec -120^\circ\end{align*}
- \begin{align*}\cot -120^\circ\end{align*}
- \begin{align*}\csc -45^\circ\end{align*}
- \begin{align*}\sec -45^\circ\end{align*}
- \begin{align*}\tan -45^\circ\end{align*}
- \begin{align*}\cos -135^\circ\end{align*}
- \begin{align*}\csc -135^\circ\end{align*}
- \begin{align*}\sec -135^\circ\end{align*}
- \begin{align*}\tan -210^\circ\end{align*}
- \begin{align*}\sin -270^\circ\end{align*}
- \begin{align*}\cot -90^\circ\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.19.

### Image Attributions

Here you'll learn how to find the results of trigonometric functions for negative angles.

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