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°.
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°.
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.
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).
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
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