Question

If $\sin(\pi \cos \theta) = \cos(\pi \sin \theta),$ then $\sin 2 \theta$ equals

• A. $\pm \frac{3}{4}$
• B. $\pm \sqrt{2}$
• C. $\pm \frac{1}{\sqrt{3}}$
• D. $\pm \frac{1}{2}$

Answer 1

Answer: (A)

$\sin\left(\pi \cos \theta\right) =\cos\left(\pi \sin\theta\right) $
$ \Rightarrow \cos \left(\frac{\pi}{2} -\pi \cos \theta\right) =\cos \left(\pi \sin \theta\right) $
$ \Rightarrow \frac{\pi}{2} - \pi \cos \theta -\pm \pi \sin \theta $
$ \Rightarrow \frac{\pi }{2} =\pi \cos \theta \pm \pi \sin \theta$
$ \frac{1 }{2} = \cos \theta \pm \sin \theta$
Squaring both sides, we get
$ \frac{1}{4}= \cos^{2} \theta + \sin^{2} \theta \pm 2 \sin \theta \cos \theta $
$\Rightarrow \frac{1}{4} = 1 \pm 2 \sin \theta \cos \theta $
$\Rightarrow \frac{1}{4} - 1 \pm \sin 2\theta$
$ \Rightarrow - \frac{3}{4} = \pm \sin 2\theta \Rightarrow \sin 2\theta \pm \frac{3}{4}$