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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(-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 r
^{2}= x^{2}+ y^{2}

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 × (square root of 3)/2 = square root of 3 - r
^{2}= x^{2}+ y^{2}= (-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 60
^{o}= sin30^{o}= 1/2 - cos(180
^{o}- 60^{o}) = -cos60^{o}= -1/2 - so, a = 180
^{o}- 60^{o}= 120^{0}

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}