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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(-2cos60o, 2sin60o). 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(-2cos60o, 2sin60o). We need to determine which quadrant the terminal side of the angle a lies in.

x = -2 cos 60o = -2 sin 30o = -2 × (1/2) = -1
y = 2 sin 60o = 2 × (square root of 3)/2 = square root of 3
r2 = x2 + y2 = (-1)2 + (square root of 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, square root of 3), which lies in quadrant II, so, cos a < 0.
since cos 60o = sin30o = 1/2
cos(180o - 60o) = -cos60o = -1/2
so, a = 180o - 60o = 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 120o