Question

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

a) Abigail is correct.
b) Abigail is incorrect.

Answer 1

Abigail is incorrect. The reference angle for -765° is not 45°. The correct values for cos(-765°) and sin(-765°) are not equal to cos(45°) and sin(45°) respectively.

Step-by-step explanation:

Abigail is incorrect. The reference angle for -765° is not 45°. A reference angle is the acute angle formed between the terminal side of the given angle and the x-axis in standard position. To find the reference angle for -765°, we can add 360° multiple times until we get an angle between 0° and 360°. In this case, -765° + 360° = -405°, which is equivalent to 315° in positive form. The reference angle for 315° is 45°.

To find cos(-765°) and sin(-765°), we use the trigonometric functions for positive angles. We know that cos(315°) = cos(45°) = √(2)/2 and sin(315°) = sin(45°) = √(2)/2. However, cos(-765°) = cos(315°) = √(2)/2 ≠ cos(45°), and sin(-765°) = sin(315°) = √(2)/2 ≠ sin(45°).