Question

Which two of the following expressions are opposite of cos(θ) on the unit circle, where θ is in radians? Choose the correct pair of expressions that are opposite of cos(θ) on the unit circle.

A) sin(θ) and −sin(θ)
B) −cos(θ) and cos(θ)
C) −sin(θ) and sin(θ)
D) cos(θ) and −cos(θ)

Answer 1

The expressions opposite of cos(θ) on the unit circle are −cos(θ) and cos(θ), reflecting the point across the origin on the coordinate plane. Sin(θ) is not related since it represents the y-coordinate, not the x-coordinate like cosine. The correct answer is D) cos(θ) and −cos(θ).

Step-by-step explanation:

The expressions that are opposite of cos(θ) on the unit circle refer to the values that would represent the cosine of an angle θ when it is reflected about the origin. Since cosine is the x-coordinate of a point on the unit circle, the opposite of cos(θ) would be −cos(θ). This is because reflecting a point across the origin in a coordinate plane inverts both the x and y coordinates, resulting in (cos(θ), sin(θ)) becoming (−cos(θ), −sin(θ)). In the context of the options provided, the correct pair that are opposite of cos(θ) are −cos(θ) and cos(θ), since sin(θ) represents the y-coordinate and is not directly concerned with the x-coordinate, which is what cosine represents on the unit circle. Therefore, the correct answer is D) cos(θ) and −cos(θ).