Question

Which of the following are solutions to the equation sinxcosx = - 1/4

Answer 1

The equation is given below as

$\sin x\cos x=-\frac{1}{4}$

Recall that

$\begin{gathered} \sin 2x=2\sin xcosx \\ \text{which means that } \\ \frac{\sin2x}{2}=\sin x\cos x \end{gathered}$

Therefore, equating the two-equation, we will have

$\frac{\sin 2x}{2}=-\frac{1}{4}$

cross multiply, we will have

$\begin{gathered} \sin 2x=-\frac{1}{4}\times2 \\ \sin 2x=-\frac{1}{2} \end{gathered}$

Therefore,

we will get the arc sign of both sides in radians to get

$\begin{gathered} 2x=\sin ^{-1}(-\frac{1}{2}) \\ x=\frac{1}{2}\sin ^{-1}(-\frac{1}{2}) \end{gathered}$

In radians, the final answer will be

$\begin{gathered} x=\frac{7\pi}{12}+n\pi \\ \text{and} \\ x=\frac{11\pi}{12}+n\pi \end{gathered}$

Therefore,

The finals answers are OPTION A and OPTION D