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 × 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 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