Question

Abigail says that since the reference angle for a rotation through - 765° has measure 459 then cos(-765°) = cos(45°), and sin(-765°) = sin(45°). Explain why she is or is not correct.

Answer 1

Abigail is correct that cos(-765°) = cos(45°) due to the even nature of the cosine function. However, she is incorrect about sin(-765°) = sin(45°) because the sine function is odd, which means sin(-765°) actually equals -sin(45°).

Step-by-step explanation:

Abigail's assertion about the cosine and sine values of the angles -765° and 45° requires us to understand the concept of reference angles and the periodic nature of trigonometric functions. A reference angle is the acute angle that a given angle makes with the x-axis.

Since a full rotation is 360°, we can simplify -765° by adding multiples of 360° until we get an angle that's between 0° and 360°. So, -765° + 2×360° = -765° + 720° = -45°, which lies in the fourth quadrant. However, the reference angle is positive, hence the reference angle for -765° is 45°.

The cosine function is even, which means that cos(θ) = cos(-θ). Therefore, cos(-765°) = cos(45°). But the sine function is odd, meaning that sin(-θ) = -sin(θ). As such, sin(-765°) is not equal to sin(45°), instead, it equals -sin(45°). So, while Abigail is correct about the cosine, she is incorrect about the sine.