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1.20: Trigonometric Functions of Angles Greater than 360 Degrees

Difficulty Level: At Grade Created by: CK-12
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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 390. As you learned previously, you can think of this angle as a full 360 degree rotation, plus an additional 30 degrees. Therefore 390 is coterminal with 30. As you saw above with negative angles, this means that 390 has the same ordered pair as 30, and so it has the same trig values. For example,

cos390=cos30=32

 

In general, if an angle whose measure is greater than

360 has a reference angle of 3045, or 60, 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 360

Find the value of the following expressions:

1. sin420

sin420=32

420 is a full rotation of 360 degrees, plus an additional 60 degrees. Therefore the angle is coterminal with 60, and so it shares the same ordered pair, (12,32). The sine value is the ycoordinate.

2. tan840

tan840=3

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

840=360+360+120

Therefore 840 is coterminal with 120, so the ordered pair is (12,32). The tangent value can be found by the following:

tan840=tan120=yx=3212=32×21=3

3. cos540

cos540=1

540 is a full rotation of 360 degrees, plus an additional 180 degrees. Therefore the angle is coterminal with 180, 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 360×3.5=1260. However, as you learned in this unit, this is equivalent to 180. So you can use that value in your computations:

sin1260=sin180=0cos1260=cos180=1

Example 2

Find the value of the expression: sin570

Since 570 has the same terminal side as 210, sin570=sin210=121=12

Example 3

Find the value of the expression: cos675

 Since 675 has the same terminal side as 315, cos675=cos315=221=22

Example 4

Find the value of the expression: sin480

Since 480 has the same terminal side as 120, sin480=sin120=321=32

Review

Find the value of each expression.

  1. sin405
  2. cos810
  3. tan630
  4. cot900
  5. csc495
  6. sec510
  7. \begin{align*}\cos 585^\circ\end{align*}
  8. \begin{align*}\sin 600^\circ\end{align*}
  9. \begin{align*}\cot 495^\circ\end{align*}
  10. \begin{align*}\tan 405^\circ\end{align*}
  11. \begin{align*}\cos 630^\circ\end{align*}
  12. \begin{align*}\sec 810^\circ\end{align*}
  13. \begin{align*}\csc 900^\circ\end{align*}
  14. \begin{align*}\tan 600^\circ\end{align*}
  15. \begin{align*}\sin 585^\circ\end{align*}
  16. \begin{align*}\tan 510^\circ\end{align*}
  17. 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. 

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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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Difficulty Level:
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Date Created:
Sep 26, 2012
Last Modified:
Mar 23, 2016
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