Sin cos - Colorado State University

Objective: Compute with Radian Measure

Compute Trigonometric Function Values

You can use the Unit Circle to compute trigonometric values by noting the signs (+ or -) of the trigonometric functions sine, cosine, and tangent.

The table below summarizes the signs for the various functions. To determine the

sign

of

tangent,

recall that

tan

=

sin cos

Function sine cosine tangent

Quadrant I

+ + +

Quadrant II

+ -

Quadrant III

+

Quadrant IV

+ -

Objective: Compute with Radian Measure

Determining Angles in Radians and Degrees

When you know a trigonometric function value you can compute inverse functions

( ) sin-1, cos-1, tan-1 to determine the angle measures in degrees 0? to 360? and in

radians from 0 to 2 . Later we will explore the graphs of inverse function, but for

now we will use what we know about reference angles and the signs of the

functions in the various quadrants. For this discussion we will focus on the values

for the reference angle , or 30? and use three examples. From the unit circle we

6

have

sin

6

=

1 2

,

cos

6

=

3 2

,

tan

6

=

3 3

i) To determine the solutions of the equation sin = - 1 , we know that the 2

reference angle is 30? or .

6

If you look at the calculator solution you

will see -30?. Although on a test, you

will be expected to know the special

Unit Circle angles without a calculator.

=

sin

-1

-

1 2

=

-30?

Sine is negative in Quadrants III and IV, so we have the solutions:

= 180? + 30? = 210? = 360? - 30? = 330?

= + = 7 66

= 2 - = 11 66

While you should be able to determine these values exactly, knowing the Unit

Circle, a quick calculator check shows we are right by looking at the sine values.

Objective: Compute with Radian Measure

ii) To determine the solutions of the equation cos = - 3 , we know that the 2

reference angle is 30? or but cosine is negative in Quadrants II and III.

6

If you look at the calculator solution you

will see the Quadrant II solution.

= cos-1 -

3 2

=

150?

= cos-1 -

3 2

=

5 6

Now to compute the reference angle we have: 180? -150? = 30?

- 5 = 66

Now to compute the other solutions using the reference angle, 30? or we have:

6

= 180? + 30? = 210?

= + = 7 66

You should be able to determine these values exactly, knowing the Unit Circle by evaluating cosine of each.

cos210 = - 3 2

cos 7 = - 3 62

Objective: Compute with Radian Measure

iii) To determine the solutions of the equation tan = 3 , we know that the 3

reference angle is 30? or .

6

If you look at the calculator solution you

will see.

= tan -1

3 3

=

30?

Tangent is positive in Quadrants I and III, so we have the solutions:

= 30? = 180? + 30? = 210?

= 6

= + = 7 66

It is possible to use reference angles to solve problems that do not involve special angles. For example, consider the following problem.

iv) Determine the solutions of the equation cos = -0.82 .

If you look at the calculator solution you will see the Quadrant II solution:

= cos-1(- 0.82) 145.08?

or

= cos-1(- 0.82) 2.53

Now to compute the reference angle we

have: 180? -145.08? 34.92?

or

34.92? 180?

0.61

Objective: Compute with Radian Measure

Cosine is negative in both Quadrants II and III. We can use the reference angle to calculate the Quadrant III solution.

180 + 34.92? 214.92?

or

214.92?

180?

3.75

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