Find the value of cos(a) and the degree measure of the angle a

Given: The terminal side of the angle a pass the point P(-2cos60^o, 2sin60^o). What is the value of cos(a)? What is the degree measure of the angle a? ( 0 < a < 2pi)

Solution:

By the definition: cos a = x/r and r² = x² + y²

in which, x is the x-coordinate of the point which lies on the terminal side of the angle a and y is the y-coordinate of the point which lies on the terminal side of the angle a. So, we need to find x, y and r.

From the given condition, the coordinate of the point that lies on the terminal side of the angle a is P(-2cos60^o, 2sin60^o). We need to determine which quadrant the terminal side of the angle a lies in.

x = -2 cos 60^o = -2 sin 30^o = -2 × (1/2) = -1

y = 2 sin 60^o = 2 × Sqrt (3)/2 = Sqrt (3)

r² = x² + y² = (-1)² + Sqrt (3)² = 1 + 3 = 4

so, r = 2 (note: r is always positive.)

cos a = x/r = -1/2

since the coordinate of the point is P(-1, Sqrt (3)), which lies in quadrant II, so, cos a < 0.

since cos 60^o = sin30^o = 1/2

cos(180^o - 60^o) = -cos60^o = -1/2

so, a = 180^o - 60^o = 120⁰

The terminal side of the angle a lies in quadrant II, the value of cos a = -1/2 and the degree measure of the angle a is 120^o