First slide

Trigonometric equations

Question

If $0 \leq x \leq π$ and $81^{\sin^{2} x} + 81^{\cos^{2} x} = 30$ , then x is equal to

Moderate

A

$\frac{π}{6}, \frac{π}{3}$
B

$\frac{π}{3}, \frac{π}{2}$
C

$\frac{5 π}{6}, \frac{π}{3}$
D

$\frac{2 π}{3}, \frac{π}{3}$

Solution

Let $81^{\sin^{2} x} = y$ . Then, $81^{\cos^{2} x} = 81^{1 - \sin^{2} x} = 81 y^{- 1}$
Now,
$\begin{array}{l} 81^{\sin^{2} x} + 81^{\cos^{2} x} = 30 \\ \Rightarrow y + \frac{81}{y} = 30 \\ \Rightarrow y^{2} - 30 y + 81 = 0 \end{array}$
$\Rightarrow y = 3$ or, $y = 27$
$\Rightarrow 81^{\sin^{2} x} = 3$ or $81^{\sin^{2} x} = 27$
$= 3^{4 \sin^{2} x} = 3^{1}$ or $3^{4 \sin^{2} x} = 3^{3}$
$\Rightarrow 4 \sin^{2} x = 1, 4 \sin^{2} x = 3$
$\Rightarrow \sin x = \pm \frac{1}{\sqrt{2}}$ or, $\sin x = \pm \frac{\sqrt{3}}{2} \Rightarrow x = \frac{π}{6}$ or $\frac{π}{3}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App