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

### Watch This

### Guidance

Consider the angle . As you learned previously, you can think of this angle as a full 360 degree rotation, plus an additional 30 degrees. Therefore is coterminal with . As you saw above with negative angles, this means that has the same ordered pair as , and so it has the same trig values. For example,

In general, if an angle whose measure is greater than has a reference angle of , , or , or if it is a quadrantal angle, we can find its ordered pair, and so we can find the values of any of the trig functions of the angle. Again, determine the reference angle first.

#### Example A

Find the value of the expression:

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Solution:
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is a full rotation of 360 degrees, plus an additional 60 degrees. Therefore the angle is coterminal with , and so it shares the same ordered pair, . The sine value is the coordinate.

#### Example B

Find the value of the expression:

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Solution:
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is two full rotations, or 720 degrees, plus an additional 120 degrees:

Therefore is coterminal with , so the ordered pair is . The tangent value can be found by the following:

#### Example C

Find the value of the expression:

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Solution:
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is a full rotation of 360 degrees, plus an additional 180 degrees. Therefore the angle is coterminal with , and the ordered pair is (-1, 0). So the cosine value is -1.

### Vocabulary

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Coterminal:
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Two angles are
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coterminal
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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:

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Solutions:
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1. Since has the same terminal side as ,

2. Since has the same terminal side as ,

3. Since has the same terminal side as ,

### Concept Problem Solution

Since you've gone around the track 3.5 times, the total angle you've traveled is . However, as you learned in this unit, this is equivalent to . So you can use that value in your computations:

### Practice

Find the value of each expression.

- Explain how to evaluate a trigonometric function for an angle greater than .