Mathematics - Trigonometric Equation Question with Solution | TestHub

$\left(\mathrm{I}\right)$$f\left(x\right)=\sin \left(\pi \cos x\right)$and$X=\left\{x:f\left(x\right)=0\right\}$
$f\left(x\right)=0$
$\Rightarrow \sin \left(\pi \cos x\right)=0$
$\Rightarrow \pi \cos x=m\pi , m\in I$
$\Rightarrow \cos x=m$
$\cos x=0, 1, -1$
$\Rightarrow x=\frac{n\pi }{2}$
$\Rightarrow X=\left\{\frac{n\pi }{2}, n\in I\right\}\Rightarrow X=\left\{\pm \frac{\pi }{2}, \pm \pi ,\pm \frac{3\pi }{2},\pm 2\pi \dots \dots \right\}$
$\left(\mathrm{I}\right)-P, Q$
$\left(\mathrm{I}\mathrm{I}\right) g\left(x\right)=\cos \left(2\pi \sin x\right)$and$Z=\left\{x:g\left(x\right)=0\right\}$
$\cos \left(2\pi \sin x\right)=0$
$\Rightarrow 2\pi \sin x=\left(2m+1\right)\frac{\pi }{2},m\in I$
$\Rightarrow \sin x=\frac{2n+1}{4}$
$\Rightarrow \sin x=-\frac{1}{4},\frac{1}{4},\frac{-3}{4},\frac{3}{4}$
$\Rightarrow Z=\left\{n\pi \pm {\sin }^{-1}\left(\frac{1}{4}\right), n\pi \pm {\sin }^{-1}\left(\frac{3}{4}\right)\right\},n\in I$
$\left(\mathrm{I}\mathrm{I}\right)-Q, T$
$\left(\mathrm{I}\mathrm{I}\mathrm{I}\right) f\left(x\right)=\sin \left(\pi \cos x\right)$and$Y=\left(x:f^{\text{'}}\left(x\right)=0\right)$
$f^{\text{'}}\left(x\right)=\cos \left(\pi \cos x\right).\left(-\pi \sin x\right)=0$
Now,$\cos \left(\pi \cos x\right)=0\Rightarrow \pi \cos x=\left(2m+1\right)\frac{\pi }{2},m\in I$
$\Rightarrow \cos x=\frac{2m+1}{2}$
$\Rightarrow \cos x=-\frac{1}{2},\frac{1}{2}\Rightarrow x=n\pi \pm \frac{\pi }{3},n\in I$
or$\sin x=0\Rightarrow x=n\pi$
Hence,$Y=\left\{n\pi , n\pi \pm \frac{\pi }{3}\right\},n\in I$
$Y=\left\{\frac{\pi }{3},\frac{2\pi }{3}, \pi ,\frac{4\pi }{3},\frac{5\pi }{3}, 2\pi \dots \right\}$
$\left(\mathrm{I}\mathrm{I}\mathrm{I}\right)\to R$
$\left(\mathrm{I}\mathrm{V}\right)$$g\left(x\right)=\cos \left(2\pi \sin x\right)$and$W=\left\{x:g^{\text{'}}\left(x\right)=0\right\}$
$g^{\text{'}}\left(x\right)=-\sin \left(2\pi \sin x\right)\left(2\pi \cos x\right)=0$
Now,$\cos x=0\Rightarrow x=\left(2m+1\right)\frac{\pi }{2},m\in I$
or$\sin \left(2\pi \sin x\right)=0$
$2\pi \sin x=m\pi ,m\in I$
$\Rightarrow \sin x=\frac{m}{2}$
$\Rightarrow \sin x=-1, -\frac{1}{2}, 0,\frac{1}{2}, 1$
$\Rightarrow x=\frac{n\pi }{2}, n\pi \pm \frac{\pi }{6},n\in I$
$W=\left\{\frac{n\pi }{2}, n\pi \pm \frac{\pi }{6}\right\}$
$W=\left\{\frac{\pi }{6},\frac{\pi }{2},\frac{5\pi }{6},\pi , \frac{7\pi }{6},\frac{3\pi }{2}\dots .\right\}$
$\left(\mathrm{I}\mathrm{V}\right)-P, R, S$