Question

Solve sin2∅=sin∅ on the interval 0≤x< 2"absmiddle" class="latex-formula"> .
a. 0,$\frac{\pi }{3}$
b. 0, $\pi$, $\frac{\pi }{3}$, $\frac{5\pi }{3}$
c. 0, $\pi$, $\frac{2\pi }{3}$,$\frac{4\pi }{3}$
d. $\frac{3\pi }{2}$, $\frac{\pi }{2}$,$\frac{\pi }{6}$, $\frac{5\pi }{6}$

Answer 1

Answer:

$\large\boxed{b.\ 0,\ \pi,\ \dfrac{\pi}{3},\ \dfrac{5\pi}{3}}$

Step-by-step explanation:

$\text{Use}\ \sin2x=2\sin x\cos x\\\\\sin2\O=\sin\O\\\\2\sin\O\cos\O=\sin\O\qquad\text{subtract}\ \sin\O\ \text{from both sides}\\\\2\sin\O\cos\O-\sin\O=0\qquad\text{distribute}\\\\\sin\O(2\cos\O-1)=0\iff\sin\O=0\ \vee\ 2\cos\O-1=0$

$\sin\O=0\iff\O=0\ \vee\ \O=\pi\\\\2\cos\O-1=0\qquad\text{add 1 to both sides}\\\\2\cos\O=1\qquad\text{divide both sides by 2}\\\\\cos\O=\dfrac{1}{2}\iff\O=\dfrac{\pi}{3}\ \vee\ \O=\dfrac{5\pi}{3}$