Question

Solve: sin2x cosx + cos 2x sin x = √3/2

Answer 1

We are given with the expression or equation sin2x cosx + cos 2x sin x = √3/2. The expression can be patterned from the trigonometric identity sin (a + b) = sin a cos B + cos A sin B. In this case, the expression is equal to sin 3x = sqrt 3 /2. using arc sign, x is equal to pi/9 

Answer 2

Answer:

$x=\frac{2\pi n}{3} +\frac{2 \pi}{9} \hspace{8}for\hspace{8}n\in Z\\\\or\\\\x=\frac{2\pi k}{3} +\frac{ \pi}{9} \hspace{8}for\hspace{8}k\in Z$

Step-by-step explanation:

Using sine sum identity:

$sin(\alpha + \beta)=sin(\alpha) cos(\beta)+cos(\alpha) sin(\beta)$

We can reduce trigonometric functions:

$sin(2x)cos(x)+cos(2x)sin(x)=sin(2x+x)=sin(3x)$

Hence:

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

Take the inverse sine of both sides:

$3x=2 \pi n+\frac{2\pi}{3} \hspace{8}for\hspace{8}n\in Z\\\\or\\\\3x=2 \pi k+\frac{\pi}{3} \hspace{8}for\hspace{8}k\in Z$

Finally, divide both sides by 3:

$x=\frac{2\pi n}{3} +\frac{2 \pi}{9} \hspace{8}for\hspace{8}n\in Z\\\\or\\\\x=\frac{2\pi k}{3} +\frac{ \pi}{9} \hspace{8}for\hspace{8}k\in Z$