Question

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

Answer 1

`x=(2\pi n)/(3) +(2 \pi)/(9) \hspace{8}for\hspace{8}n\in Z\\\\or\\\\x=(2\pi k)/(3) +( \pi)/(9) \hspace{8}for\hspace{8}k\in Z`

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)=(√(3) )/(2)`

Take the inverse sine of both sides:

`3x=2 \pi n+(2\pi)/(3) \hspace{8}for\hspace{8}n\in Z\\\\or\\\\3x=2 \pi k+(\pi)/(3) \hspace{8}for\hspace{8}k\in Z`

Finally, divide both sides by 3:

`x=(2\pi n)/(3) +(2 \pi)/(9) \hspace{8}for\hspace{8}n\in Z\\\\or\\\\x=(2\pi k)/(3) +( \pi)/(9) \hspace{8}for\hspace{8}k\in Z`

Answer 2

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