Question

Trignometry !!!

$\sin \left(x\right)+\sin \left(\frac{x}{2}\right)=0,\:0\le \:x\le \:2\pi$

find radians

Answer 1

Answer:

oho

Explanation:

$\sin \left(x\right)+\sin \left(\frac{x}{2}\right)=0,\:0\le \:x\le \:2\pi$

find radians

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\bf{\sin \left(x\right)+\sin \left(\dfrac{x}{2}\right)=0,\:0\le \:\:x\le \:\:2\pi }$

♣ ᴛᴏ ꜰɪɴᴅ :

ʀᴀᴅɪᴀɴꜱ

♣ ᴀɴꜱᴡᴇʀ :

$\bf{\sin \left(x\right)+\sin \left(\dfrac{x}{2}\right)=0,\:0\le \:x\le \:2\pi \quad :\quad \begin{bmatrix}\mathrm{Radians:}\:&\:x=2\pi ,\:x=0,\:x=\dfrac{4\pi }{3}\:\\ \:\mathrm{Degrees:}&\:x=360^{\circ \:},\:x=0,\:x=240^{\circ \:}\end{bmatrix}}$

$\mathrm{Use\:the\:following\:identity}:\quad \sin \left(s\right)+\sin \left(t\right)=2\cos \left(\frac{s-t}{2}\right)\sin \left(\frac{s+t}{2}\right)$

$2\cos \left(\frac{-\frac{x}{2}+x}{2}\right)\sin \left(\frac{\frac{x}{2}+x}{2}\right)=0$

$\bf{Simplify\:\: 2 \cos \left(\frac{-\frac{x}{2}+x}{2}\right) \sin \left(\frac{\frac{x}{2}+x}{2}\right): 2 \cos \left(\frac{x}{4}\right) \sin \left(\frac{3 x}{4}\right) }$

$2 \cos \left(\frac{-\frac{x}{2}+x}{2}\right) \sin \left(\frac{\frac{x}{2}+x}{2}\right)$

$\frac{-\frac{x}{2}+x}{2}=\frac{x}{4}$

$\quad \frac{x}{2}+x=\frac{3 x}{4}$

$=2 \cos \left(\frac{x}{4}\right) \sin \left(\frac{3 x}{4}\right)$

$2\cos \left(\frac{x}{4}\right)\sin \left(\frac{3x}{4}\right)=0$

$\mathrm{Solving\:each\:part\:separately}$

$\cos \left(\frac{x}{4}\right)=0\quad \mathrm{or}\quad \sin \left(\frac{3x}{4}\right)=0$

$\cos \left(\dfrac{x}{4}\right)=0,0 \leq x \leq 2 \pi \quad: \quad x=2 \pi$

$\sin \left(\frac{3 x}{4}\right)=0,0 \leq x \leq 2 \pi \quad: \quad x=0, x=\frac{4 \pi}{3}$

Combine all the solutions

$x=2 \pi, x=0, x=\frac{4 \pi}{3}$