Trigonometric Functions — MATHS STD 11 Science — Question

4. None of these

Solution:

Given equation: $\text{e}^{\sin\text{x}}-\text{e}^{-\sin\text{x}}-4=0,$

Let:

$\text{e}^{\sin\text{x}}=\text{y}$

Now,

$\text{y}-\text{y}^{-1}-4=0$

$\Rightarrow\text{y}^2-4\text{y}-1=0$

$\therefore\text{y}=\frac{4\pm\sqrt{16+4}}{2}$

$\Rightarrow\text{y}=\frac{4\pm\sqrt{20}}{2}$

$\Rightarrow\text{y}=\frac{4\pm2\sqrt{5}}{2}=2\pm\sqrt{5}$

And

$\text{y}=\text{e}^{\sin\text{x}}$

$\Rightarrow\text{y}^{\sin\text{x}}={2\pm\sqrt{5}}$

Taking log on both sides, we get:

$\sin\text{x}=\log_\text{e}\big(2\pm\sqrt{5}\big)$

$\Rightarrow\sin\text{x}=\log_{e}\big(2+\sqrt{5}\big)$ or $\sin\text{x}=\log_\text{e}\big(2-\sqrt{5}\big)$

$\Rightarrow\sin\text{x}=\log_{e}\big(4.24\big)$ or $\sin\text{x}=\log_\text{e}\big(-0.24\big)$

$\log(4.24)>1$ and $\sin\text{x}$ cannot be greater than 1.

In the other case, the log of negative term occurs, which is not defined.