Question

Sinxcosx = 1/2cosx

Solve for x.

Answer 1

`x=(1)/(2) \pi + 2\pi n`

`x=(3)/(2)\pi +2 \pi n`

`x=(1)/(6)\pi+2 \pi n`

`x=(5)/(6)\pi+2 \pi n`

Explanation:

Given trigonometric equation:

`\sin x \cos x &= (1)/(2) \cos x`

Rearrange the equation so that it equals zero:

`\begin{aligned}\sin x \cos x &= (1)/(2) \cos x\\2 \sin x \cos x &= \cos x\\2 \sin x \cos x - \cos x &=0\\ \cos x (2 \sin x - 1)&=0\end{aligned}`

Apply the zero-product property.

Case 1

`\begin{aligned}\cos x &=0\\x&=\cos^(-1)(0)\\x&=(1)/(2) \pi +2\pi n, \; (3)/(2)\pi +2 \pi n\end{aligned}`

Case 2

`\begin{aligned}2\sin x - 1& = 0\\2 \sin x & = 1\\\sin x & = (1)/(2)\\x&=\sin^(-1)\left((1)/(2)\right)\\x & = (1)/(6) \pi+2 \pi n, \; (5)/(6) \pi +2 \pi n\end{aligned}`