When regarding a triangle, why is 2cosx greater than or equal to -2 (if that makes any sense)

Question

The question is from HSC Mathematics 2010 question 10 a iv, and when looking at the sample answers the first line is 2cosx is greater than or equal to -2 and I’m unsure as to how they came to that, some clarification would be greatly appreciated!

Answer 1

This follows from one the definitions of cosine. Let P=(x,y) be a point on the circle of radius 1 centered at the origin O. Let θ be the angle from the x-axis to the segment OP, The x=cos(θ) and y=sin(θ). Since the radius of the circle is 1 we get that -1≤cos(θ)≤1 and -1≤sin(θ))≤1. Multiplying through by 2 gives a lower bound of -2≤2cos(θ).