1.19: 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
At the completion of this Concept, you'll be able to calculate the values of trig functions for negative angles, and find the value of cosine for the
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Coterminal and Negative Angles
Guidance
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
Example A
Find the value of the expression:
Solution:
Example B
Find the value of the expression:
Solution:
The angle
Example C
Find the value of the expression:
Solution:
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.
Vocabulary
Negative Angle: A negative angle is an angle measured by rotating clockwise (instead of counterclockwise) from the positive 'x' axis.
Guided Practice
1. Find the value of the expression:
2. Find the value of the expression:
3. Find the value of the expression:
Solutions:
1. The angle
2. The angle
3. The angle
Concept Problem Solution
What you want to find is the value of the expression:
Solution:
Practice
Calculate each value.

sin−120∘ 
cos−120∘ 
tan−120∘  \begin{align*}\csc 120^\circ\end{align*}
 \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*}
Image Attributions
Here you'll learn how to find the results of trigonometric functions for negative angles.