Question

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

A
300
B
600
C
1200
D
1500

Solution

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

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