Question

find all solutions of equation

1. \\(\sin x = \frac{\sqrt{3}}{2}\\)
2. \\(\tan x = \sqrt{3}\\)

solve equation, give a general formula for all solutions
\\(2\cos x - \sqrt{2} = 0\\)

Explanation:

Solve the first equation

$$ \sin x = \frac{\sqrt{3}}{2} $$

$$ x = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad x = \frac{2\pi}{3} + 2k\pi, \quad k \in \mathbb{Z} $$

Solve the second equation

$$ \tan x = \sqrt{3} $$

$$ x = \frac{\pi}{3} + k\pi, \quad k \in \mathbb{Z} $$

Solve the third equation

$$ 2\cos x - \sqrt{2} = 0 \implies \cos x = \frac{\sqrt{2}}{2} $$

$$ x = \pm\frac{\pi}{4} + 2k\pi, \quad k \in \mathbb{Z} $$

Answer:

Question 1

$$x = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad x = \frac{2\pi}{3} + 2k\pi, \quad k \in \mathbb{Z}$$

Question 2

$$x = \frac{\pi}{3} + k\pi, \quad k \in \mathbb{Z}$$

Question 3

$$x = \pm\frac{\pi}{4} + 2k\pi, \quad k \in \mathbb{Z}$$