<meta http-equiv="refresh" content="1; url=/nojavascript/"> Trigonometric Functions of Angles Greater than 360 Degrees | CK-12 Foundation
Dismiss
Skip Navigation
You are reading an older version of this FlexBook® textbook: CK-12 Trigonometry Concepts Go to the latest version.

1.20: Trigonometric Functions of Angles Greater than 360 Degrees

Created by: CK-12
%
Best Score
Practice Trigonometric Functions of Angles Greater than 360 Degrees
Practice
Best Score
%
Practice Now

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 360^\circ

Watch This

The Unit Circle

Guidance

Consider the angle 390^\circ. As you learned previously, you can think of this angle as a full 360 degree rotation, plus an additional 30 degrees. Therefore 390^\circ is coterminal with 30^\circ. As you saw above with negative angles, this means that 390^\circ has the same ordered pair as 30^\circ, and so it has the same trig values. For example,

\cos 390^\circ = \cos 30^\circ = \frac{\sqrt{3}}{2}

In general, if an angle whose measure is greater than 360^\circ has a reference angle of 30^\circ, 45^\circ, or 60^\circ, 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: \sin 420^\circ

Solution:

\sin 420^\circ = \frac{\sqrt{3}}{2}

420^\circ is a full rotation of 360 degrees, plus an additional 60 degrees. Therefore the angle is coterminal with 60^\circ, and so it shares the same ordered pair, \left ( \frac{1}{2}, \frac{\sqrt{3}}{2} \right ). The sine value is the y-coordinate.

Example B

Find the value of the expression: \tan 840^\circ

Solution:

\tan 840^\circ = -\sqrt{3}

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

840 = 360 + 360 + 120

Therefore 840^\circ is coterminal with 120^\circ, so the ordered pair is \left ( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right ). The tangent value can be found by the following:

\tan 840^\circ = \tan 120^\circ = \frac{y}{x} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \frac{\sqrt{3}}{2} \times -\frac{2}{1} = -\sqrt{3}

Example C

Find the value of the expression: \cos 540^\circ

Solution:

\cos 540^\circ = -1

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

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: \sin 570^\circ

2. Find the value of the expression: \cos 675^\circ

3. Find the value of the expression: \sin 480^\circ

Solutions:

1. Since 570^\circ has the same terminal side as 210^\circ, \sin 570^\circ = \sin 210^\circ = \frac{\frac{-1}{2}}{1} = \frac{-1}{2}

2. Since 675^\circ has the same terminal side as 315^\circ, \cos 675^\circ = \cos 315^\circ = \frac{\frac{\sqrt{2}}{2}}{1} = \frac{\sqrt{2}}{2}

3. Since 480^\circ has the same terminal side as 120^\circ, \sin 480^\circ = \sin 120^\circ = \frac{\frac{\sqrt{3}}{2}}{1} = \frac{\sqrt{3}}{2}

Concept Problem Solution

Since you've gone around the track 3.5 times, the total angle you've traveled is 360^\circ \times 3.5 = 1260^\circ. However, as you learned in this unit, this is equivalent to 180^\circ. So you can use that value in your computations:

\sin 1260^\circ = \sin 180^\circ = 0\\\cos 1260^\circ = \cos 180^\circ = -1\\

Practice

Find the value of each expression.

  1. \sin 405^\circ
  2. \cos 810^\circ
  3. \tan 630^\circ
  4. \cot 900^\circ
  5. \csc 495^\circ
  6. \sec 510^\circ
  7. \cos 585^\circ
  8. \sin 600^\circ
  9. \cot 495^\circ
  10. \tan 405^\circ
  11. \cos 630^\circ
  12. \sec 810^\circ
  13. \csc 900^\circ
  14. \tan 600^\circ
  15. \sin 585^\circ
  16. \tan 510^\circ
  17. Explain how to evaluate a trigonometric function for an angle greater than 360^\circ.

Image Attributions

Description

Difficulty Level:

At Grade

Categories:

Grades:

Date Created:

Sep 26, 2012

Last Modified:

Jan 23, 2014
Files can only be attached to the latest version of Modality

Reviews

Please wait...
You need to be signed in to perform this action. Please sign-in and try again.
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.TRG.176.L.1

Original text