if θ = 3π/4, find the exact value of each expression below. (a) cos(-θ)= (b) cos²θ = (c) cos 2θ =

Explanation:

Step1: Recall cosine - even property

The cosine function is even, i.e., $\cos(-\theta)=\cos\theta$. Given $\theta = \frac{3\pi}{4}$, and $\cos(\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}$, so $\cos(-\theta)=\cos(\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}$.

Step2: Calculate $\cos^{2}\theta$

We know $\cos\theta =-\frac{\sqrt{2}}{2}$. Then $\cos^{2}\theta=\left(-\frac{\sqrt{2}}{2}\right)^{2}=\frac{2}{4}=\frac{1}{2}$.

Step3: Use double - angle formula

The double - angle formula for cosine is $\cos2\theta = 2\cos^{2}\theta - 1$. Since $\cos^{2}\theta=\frac{1}{2}$, then $\cos2\theta=2\times\frac{1}{2}-1 = 0$.

Answer:

(a) $-\frac{\sqrt{2}}{2}$ (b) $\frac{1}{2}$ (c) $0$