Find all x so that sine and cosine have specified values

Find all $x \in \mathbb{R}$ to satisfy the equations in each of the following parts.

$\sin x = 1$ .
$\cos x = 1$ .
$\sin x = -1$ .
$\cos x = -1$ .

We use the result of the previous exercise (Section 2.8, Exercise 1), and the fundamental properties of the sine and cosine given in section 2.5-2.7 of Apostol in the following.

$\sin x = 1 \implies x = \frac{\pi}{2} + 2n \pi$ , using fundamental property 2 ( $\sin \frac{\pi}{2} = 1$ ), and the $2 \pi$ -periodicity of sine.
$\cos x = 1 \implies x = 2n \pi$ , using Theorem 2.3, part (d) (Apostol, Section 2.5).
$\sin x = -1 \implies -\sin x = 1 \implies \sin (-x) = 1 \implies -x = \frac{\pi}{2} + 2n pi$ using part (a). So, $x = \frac{3 \pi}{2} + 2n \pi$ .
$\cos x = -1 \implies x = (2n+1)\pi$ using fundamental property 2 ( $\cos \pi = -1$ ) and the $2 \pi$ -periodicity of the cosine function.