Question

Find
x
for
e to the power sin x - e to the power -sin x =4​

Answer 1

Find

x

for

e to the power sin x - e to the power -sin x =4​

$\textbf{Given:}$

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

$\textbf{To find:}$

$\text{The value of x}$

$\textbf{Solution:}$

$\text{Consider,}$

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

$e^{sin\,x}-\dfrac{1}{e^{sin\,x}}=4$

$\dfrac{(e^{sin\,x})^2-1}{e^{sin\,x}}=4$

$(e^{sin\,x})^2-1=4\,e^{sin\,x}$

$(e^{sin\,x})^2-4\,e^{sin\,x}-1=0$

$t^2-4\,t-1=0$

$t=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

$t=\dfrac{4\pm\sqrt{(-4)^2-4(1)(-1)}}{2(1)}$

$t=\dfrac{4\pm\sqrt{16+4}}{2}$

$t=\dfrac{4\pm\sqrt{4{\times}5}}{2}$

$t=\dfrac{4\pm\,2\sqrt{5}}{2}$

$t=2\pm\sqrt{5}$

$\textbf{Case(i):}$

$t=2+\sqrt{5}$

$e^{sin\,x}=2+\sqrt{5}$

$\implies\,sin\,x=\log({2+\sqrt{5}})$

$\implies\,x=sin^{-1}[\log({2+\sqrt{5}})]$

$\textbf{Case(iI):}$

$t=2-\sqrt{5}$

$e^{sin\,x}=2-\sqrt{5}$

$\textbf{This is not possible because e^x cannot be negative}$

$\textbf{Answer:}$

$\textbf{The value of x is \bf\,sin^{-1}[\log({2+\sqrt{5}})] }$

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