# EVALUATING TRIGONOMETRIC FUNCTIONS FOR ANY ANGLE

To find the values of the trigonometric functions for any angle u, we carry out the following steps.

Step 1 :

Find the reference angle B associated with the angle A.

Step 2 :

Determine the sign of the trigonometric function of A by noting the quadrant in which A lies.

Step 3 :

The value of the trigonometric function of A is the same, except possibly for sign, as the value of the trigonometric function of B.

Example 1 :

Using the reference angle to evaluate sin 240°.

Solution :

The angle 240° has its terminal side in quadrant III, as shown in figure below. The reference angle is therefore

240° - 180°  =  60°,

and the value of sin 240° is negative. Thus

sin 240°  =  -sin 60°  =  √3/2

Example 2 :

Using the reference angle to evaluate cot 495°.

Solution :

The angle 495° is coterminal with the angle 135°and the terminal side of this angle is in quadrant II, as shown in figure below. So the reference angle is

180° - 135°  =  45°,

and the value of cot 495° is negative. We have

cot 495°  =  cot 135°  =  -cot 45°  =  -1

Example 3 :

Using the reference angle to evaluate sin 16π/3.

Solution :

The angle 16π/3 is coterminal with 4π/3, and these angles are in quadrant III, as shown in the figure below. Thus, the reference angle is

4π/3 - π  =  π/3

Because the value of sine is negative in quadrant III, we have

sin 16π/3  =  sin 4π/3  =  -sin π/3  =  -√3/2

Example 4 :

Using the reference angle to evaluate sec (-π/4).

Solution :

The angle -π/4 is in quadrant IV, and its reference angle is π/4, as shown in the figure below. Because secant is positive in this quadrant, we get

sec (-π/4)  =  +sec (π/4)  =  √2/2 Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

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