Question

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

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

Answer 1

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

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

Given $0\le \theta \le \pi$

${81}^{{\sin }^2\theta }+{81}^{{\cos }^2\theta }=30$

$\Rightarrow {81}^{{\sin }^2\theta }+{81}^{1-{\sin }^2\theta }=30$

$[\because {\sin }^2\theta +{\cos }^2\theta =1]$

$\Rightarrow {81}^{{\sin }^2\theta }+\frac{81}{{81}^{{\sin }^2\theta }}=30$

Let ${81}^{{\sin }^2\theta }=t$

$\Rightarrow t+\frac{81}{t}=30$

$\frac{t^2+81}{t}=30$

$t^2+81+30t=0$

$t^2-30t+81=0$

$t^2-27t-3t+81=0$

$t(t-27)-3(t-27)=0$

$(t-27)(t-3)=0$

$t=27$ or $3$

$\Rightarrow {81}^{{\sin }^2\theta }=27$ or $3$

${81}^{{\sin }^2\theta }=27$ ${81}^{{\sin }^2\theta }=3$

$(3^4{)}^{{\sin }^{2}x}=3^3$ $3^{4{\sin }^{2}x}=3^1$

$3^{4{\sin }^{2}x}=3^3$ $4{\sin }^{2}x=1$

$4{\sin }^{2}x=3$ ${\sin }^{2}x=\frac{1}{4}$

${\sin }^{2}x=\frac{3}{4}$ $\sin x=\pm \frac{1}{2}\Rightarrow \sin x=\frac{1}{2}$

$\Rightarrow x={30}^o$ or ${150}^o$

$[\because 0\le \theta \le \pi ]$

$\sin x=\pm \frac{\sqrt{3}}{2}\Rightarrow \sin x=\frac{\sqrt{3}}{2}[\because 0\le \theta \le \pi \Rightarrow sineispositive]$

$\Rightarrow x={60}^o$ or ${120}^o[\because 0\le \theta \le \pi ]$