Answer
For $\theta$ near $\pi / 4$ ,
$$
f(\theta) \approx \sin \frac{\pi}{4}+\left(\cos \frac{\pi}{4}\right)\left(\theta-\frac{\pi}{4}\right)
$$
For $\theta$ near 3$\pi / 4$ ,
$$
f(\theta) \approx \sin \frac{3 \pi}{4}+\left(\cos \frac{3 \pi}{4}\right)\left(\theta-\frac{3 \pi}{4}\right)
$$
Work Step by Step
For $\theta$ near $\pi / 4$ ,
$$
f(\theta) \approx \sin \frac{\pi}{4}+\left(\cos \frac{\pi}{4}\right)\left(\theta-\frac{\pi}{4}\right)
$$
For $\theta$ near 3$\pi / 4$ ,
$$
f(\theta) \approx \sin \frac{3 \pi}{4}+\left(\cos \frac{3 \pi}{4}\right)\left(\theta-\frac{3 \pi}{4}\right)
$$
