Question

If the period of the function $f(x)=\sin 5 x \cos 3 x$ is $\alpha$, then $\cos \alpha=$

• A. $1$
• B. $\frac{1}{\sqrt{2}}$
• C. $-\frac{1}{2}$
• D. $-1$

Answer 1

Answer: (D)

We have, $f(x)=\sin 5 x \cdot \cos 3 x$.
$=\frac{1}{2}(2 \sin 5 x \cdot \cos 3 x)-\frac{1}{2}(\sin 8 x+\sin 2 x)$
Clearly, $\sin 8 x$ is periodic with period $\frac{2 \pi}{8}=\frac{\pi}{4}$ and
$\sin 2 x$ is periodic with period $\frac{2 \pi}{2}=\pi$
$\therefore $ Period of $f(x)=\text { L.C.M }\left\{\frac{\pi}{4}, \pi\right\}$
$\frac{\text { L.C. } M\{\pi, \pi\}}{\text { H.C.F }\{4,1\}}=\frac{\pi}{1}=\pi \Rightarrow \alpha=\pi$
Hence, $\cos \alpha=\cos \pi=-1$