Some Basic Trigonometry

Geometry Level 3

$\large{ e }^{ \sin { x } }-{ e }^{ -\sin { x } }-4=0$

What are the real solutions of the equation above?

x=0

x\sin ^{ -1 }{ [\log { (2-\sqrt { 5 } ) } ] }

none of these no real solution

1 solution

Swagat Panda
Jun 23, 2015

$\text{Put } { e }^{ \sin { x } }=t \text{ and the given equation becomes } { t }^{ 2 }-4t-1=0\\\Rightarrow { e }^{ \sin { x } }=2\pm \sqrt { 5 }$ .

$\text{Now since } 2-\sqrt { 5 } \text{ is negative and is log is not defined for negative values.}$ $\Rightarrow \sin { x } >1 \text{ which is not possible. Hence there are no real solutions for this equation.}$

There's an even easier method. The maximum of $e^{y}-e^{-y}$ occurs at $y = 1$ when $y \in [-1, 1]$ . Since $e-e^{-1} < e < 4$ , we have no real solutions.

Jake Lai - 5 years, 11 months ago

there's an even easier method just look at the options and you can come to the conclusion that there is no real solution

Gokul Kumar - 5 years, 11 months ago

Some Basic Trigonometry

1 solution

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