Question

If $0\le \theta \le \pi$ and ${81}^{{\sin }^2\theta }+{81}^{{\cos }^2\theta }=30$, then $\theta$

• A. ${30}^0$
• B. ${60}^0$
• C. ${120}^0$
• D. ${150}^0$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A ${60}^0$
B ${120}^0$
C ${30}^0$
D ${150}^0$

${81}^{{\sin }^2\theta }+{81}^{{\cos }^2\theta }=30$
$\Rightarrow {81}^{{\sin }^2\theta }+{81}^{1-{\sin }^2\theta }=30$
Assume ${81}^{{\sin }^2\theta }=x$
$\Rightarrow x+\frac{81}{x}=30$
$\Rightarrow x^2-30x+81=0$
$\Rightarrow (x-3)(x-27)=0$
$\Rightarrow x=3,27\Rightarrow {81}^{{\sin }^2\theta }=3^{4{\sin }^2\theta }=3^1,3^3$
$\Rightarrow {\sin }^2\theta =\frac{1}{4},\frac{3}{4}$
given $0\le \theta \le \pi$,
therefore, $\sin \theta =\frac{1}{2},\frac{\sqrt{3}}{2}$
Hence,
$\theta ={30}^{\circ },{60}^{\circ },{120}^{\circ },{150}^{\circ },$
Hence, options 'A', 'B', 'C' and 'D' are correct.