Question

sinx +tanx/2 = 0 .find general solution

Answer 1

sinx +tanx/2 = 0 .find general solution

Solution.

Substituting $\tan x = \frac{\sin x}{\cos x}$ and factorizing the left part we have

$$\sin x + \frac {\sin x}{2 \cos x} = 0 \quad \text {or} \quad \frac {\sin x (2 \cos x + 1)}{2 \cos x} = 0.$$

From here

$$\cos x \neq 0 \quad \text {and} \quad \sin x = 0 \quad \text {or} \quad 2 \cos x + 1 = 0.$$

1) $\sin x = 0 \Rightarrow x = \pi k, k \in \mathbb{Z}$ .

2) $2\cos x + 1 = 0 \Rightarrow \cos x = \frac{-1}{2} \Rightarrow x = \pm \arccos \left(\frac{-1}{2}\right) + 2\pi k \Rightarrow x = \pm (\pi - \arccos \frac{1}{2}) + 2\pi k \Rightarrow x = \pm \frac{2\pi}{3} + 2\pi k, k \in \mathbb{Z}$ .

Answer: $x = \pi k, x = \pm \frac{2\pi}{3} + 2\pi k, k \in \mathbb{Z}$ .