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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.

  1. \sin x = 1.
  2. \cos x = 1.
  3. \sin x = -1.
  4. \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.

  1. \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.
  2. \cos x = 1 \implies x = 2n \pi, using Theorem 2.3, part (d) (Apostol, Section 2.5).
  3. \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.
  4. \cos x = -1 \implies x = (2n+1)\pi using fundamental property 2 (\cos \pi = -1) and the 2 \pi-periodicity of the cosine function.

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