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Trigonometric Functions of Angles Greater than 360 Degrees

Based on coterminal and reference angles.

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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?

Angles Greater Than 360°

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.

Let's look at some problems involving angles greater than 

Find the value of the following expressions:

1. 

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.

2. 

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:

3. 

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.

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 . However, as you learned in this unit, this is equivalent to . So you can use that value in your computations:

Example 2

Find the value of the expression:

Since has the same terminal side as ,

Example 3

Find the value of the expression:

 Since has the same terminal side as ,

Example 4

Find the value of the expression:

Since has the same terminal side as ,

Review

Find the value of each expression.

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

Review (Answers)

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

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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.

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