Q60 / 160 · TS EAMCET 2018 (05 May Shift 2) · Mathematics

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
    $-\frac{1}{2}$
  • Correct answer
    D
    -1

Solution

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 $f(x)=$ L.C.M $\left\{\frac{\pi}{4}, \pi\right\}$ $$ \frac{\operatorname{L.C} M\{\pi, \pi\}}{\operatorname{H.CF}\{4,1\}}=\frac{\pi}{1}=\pi \Rightarrow \alpha=\pi $$ Hence, $\cos \alpha=\cos \pi=-1$

Hence, the correct answer is option: 'D'.

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