1.20: Trigonometric Functions of Angles Greater than 360 Degrees
While out at the local amusement park with friends, you take a ride on the Go Karts. You ride around a circular track in the carts three and a half times, and then stop at a "pit stop" to rest. While waiting for your Go Kart to get more fuel, you are talking with your friends about the ride. You know that one way of measuring how far something has gone around a circle (or the trig values associated with it) is to use angles. However, you've gone more than one complete circle around the track.
Is it still possible to find out what the values of sine and cosine are for the change in angle you've made?
When you complete this Concept, you'll be able to answer this question by computing the trig values for angles greater than
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Guidance
Consider the angle
In general, if an angle whose measure is greater than
Example A
Find the value of the expression:
Solution:
Example B
Find the value of the expression:
Solution:
Therefore
Example C
Find the value of the expression:
Solution:
Vocabulary
Coterminal: Two angles are coterminal if they are drawn in the standard position and both have terminal sides that are at the same location.
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. Since
2. Since
3. Since
Concept Problem Solution
Since you've gone around the track 3.5 times, the total angle you've traveled is
Practice
Find the value of each expression.

sin405∘ 
cos810∘ 
tan630∘ 
cot900∘ 
csc495∘ 
sec510∘  \begin{align*}\cos 585^\circ\end{align*}
 \begin{align*}\sin 600^\circ\end{align*}
 \begin{align*}\cot 495^\circ\end{align*}
 \begin{align*}\tan 405^\circ\end{align*}
 \begin{align*}\cos 630^\circ\end{align*}
 \begin{align*}\sec 810^\circ\end{align*}
 \begin{align*}\csc 900^\circ\end{align*}
 \begin{align*}\tan 600^\circ\end{align*}
 \begin{align*}\sin 585^\circ\end{align*}
 \begin{align*}\tan 510^\circ\end{align*}
 Explain how to evaluate a trigonometric function for an angle greater than \begin{align*}360^\circ\end{align*}.
Image Attributions
Here you'll learn how to find the values of trigonometric functions for angles exceeding 360 degrees.