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?

### Angles Greater Than 360°

Consider the angle

In general, if an angle whose measure is greater than

Let's look at some problems involving angles greater than

Find the value of the following expressions:

1.

2.

Therefore

3.

### Examples

#### Example 1

Earlier, you were asked if it is still possible to find out what the values of sine and cosine are for the change in angle.

Since you've gone around the track 3.5 times, the total angle you've traveled is

#### Example 2

Find the value of the expression:

Since

#### Example 3

Find the value of the expression:

Since

#### Example 4

Find the value of the expression:

Since

### Review

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*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.20.