use reference angles to find the exact value of the expression cos 585°. do not use a calculator. cos 585° =…

Explanation:

Step1: Reduce the angle

Subtract 360° from 585° to get an equivalent angle in one - full rotation. $585^{\circ}-360^{\circ}=225^{\circ}$.

Step2: Determine the reference angle

The angle $225^{\circ}$ is in the third - quadrant. The reference angle $\theta_{r}$ for an angle $\theta = 225^{\circ}$ is $\theta_{r}=225^{\circ}-180^{\circ}=45^{\circ}$.

Step3: Find the cosine value

In the third - quadrant, cosine is negative. And $\cos45^{\circ}=\frac{\sqrt{2}}{2}$, so $\cos225^{\circ}=-\cos45^{\circ}=-\frac{\sqrt{2}}{2}$.

Answer:

$-\frac{\sqrt{2}}{2}$