Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# If E Sin X − E − Sin X − 4 = 0 , Then X = - Mathematics

MCQ
Sum

If $e^{\sin x} - e^{- \sin x} - 4 = 0$, then x =

#### Options

• 0

• $\sin^{- 1} \left\{ \log_e \left( 2 - \sqrt{5} \right) \right\}$

• 1

• none of these

#### Solution

none of these
Given equation:
$e^{\sin x} - e^{- \sin x} - 4 = 0$
Let :
$e^{\sin x }= y$
Now,
$y - y^{- 1} - 4 = 0$
$\Rightarrow y^2 - 4y - 1 = 0$

∴ $y = \frac{4 \pm \sqrt{16 + 4}}{2}$
$\Rightarrow y = \frac{4 \pm \sqrt{20}}{2}$
$\Rightarrow y = \frac{4 \pm 2\sqrt{5}}{2} = 2 \pm \sqrt{5}$
and,
$y = e^{\sin x}$
$\Rightarrow e^{\sin x} = 2 \pm \sqrt{5}$
Taking log on both sides, we get:
$\sin x = \log_e (2 \pm \sqrt{5})$
$\Rightarrow \sin x = \log_e ( 2 + \sqrt{5})\text{ or }\sin x = \log_e ( 2 - \sqrt{5})$
$\Rightarrow \sin x = \log_e ( 4 . 24)\text{ or }\sin x = \log_e ( - 0 . 24)$
$\log_e ( 4 . 24) > 1\text{ and }\sin x\text{ cannot be greater than }1 .$
In the other case, the log of negative term occurs, which is not defined.
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 11 Trigonometric equations
Q 15 | Page 27