Writing answers to trigonometric equation

Question

I wonder how to write answers to trigonometric equations in more elegant form. For instance if we have $ \displaystyle \sin x = \frac{\sqrt{2}}{2} \vee \sin x=-\frac{\sqrt{2}}{2}$ then I write four cases instead of just one where $\displaystyle x=\frac{\pi}{4}+\frac{k\pi}{2}$

Can anyone explain how to obtain such forms ?

Answer 1

$$\sin x=-\frac{\sqrt2}2=-\frac1{\sqrt2}=\sin\left(-\frac\pi4\right)$$

$$\implies x=n\pi+(-1)^n\left(-\frac\pi4\right)$$ where $n$ is any integer

for $\displaystyle n=2m\implies x=2m\pi-\frac\pi4$

for $\displaystyle n=2m+1\implies x=(2m+1)\pi+\frac\pi4=2m\pi+\frac{5\pi}4$

Similarly, $\displaystyle\sin x=\frac{\sqrt2}2\implies $

for $\displaystyle n=2m\implies x=2m\pi+\frac\pi4$

for $\displaystyle n=2m+1\implies x=(2m+1)\pi-\frac\pi4=2m\pi+\frac{3\pi}4$

Observe that the values can be merged into $$n\cdot\frac\pi2+\frac\pi4$$ i.e., each of the four Quadrant has the offset $\dfrac\pi4$

Answer 2

$$\sin x=\pm\frac{\sqrt2}2\iff\sin^2x=\frac24=\frac12$$

$$\iff\cos2x=1-2\sin^2x=0=\cos\frac\pi2$$

So, $2x$ must be odd multiple of $\dfrac\pi2$