Q97 / 160 · AP EAMCET 2022 (07 Jul Shift 2) · Mathematics

A ray makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with $Y$ and $Z$-axes respectively. Then, the value of the sine of the angle made by the ray with $X$-axis is

  • Correct answer
    A
    $\frac{\sqrt{3}}{2}$
  • B
    $\frac{1}{2}$
  • C
    $\frac{1}{\sqrt{2}}$
  • D
    $1$

Solution

If $\alpha, \beta, \gamma$ are angles made by a ray with $X, Y$ and $Z$-axes respectively, then $\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1$ $$ \begin{aligned} & \therefore \cos ^2 \alpha+\cos ^2\left(\frac{\pi}{3}\right)+\cos ^2\left(\frac{\pi}{4}\right)=1 \\ & \Rightarrow \cos ^2 \alpha=1-\frac{1}{4}-\frac{1}{2}=\frac{1}{4} \\ & \Rightarrow \cos \alpha= \pm \frac{1}{2} \\ & \therefore \sin \alpha=\sqrt{1-\frac{1}{4}}=\frac{\sqrt{3}}{2} \end{aligned} $$

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

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