Question

The amplitude of $e^{e^{-i\theta }}$ is equal to

• A. $\sin \theta$
• B. $-\sin \theta$
• C. $e^{\cos \theta }$
• D. $e^{\sin \theta }$

Answer 1

The correct option is C

$e^{\cos \theta }$

Step1: Given data:

The given equation is $e^{e^{-i\theta }}$.

Step2: Formula Used

According to Euler, $e^{\pm i\theta }=\cos \theta \pm i\sin \theta$.

For convenience $e^{-i\theta }=\cos \theta -i\sin \theta$ can be written as $z=r{e}^{-i\theta }----\left(1\right)$

Where $r$ is amplitude.

Step3: Simplify the given expression

$\begin{array}{l}z=e^{e^{-i\theta }}\\ z=e^{\cos \theta -i\sin \theta }\\ z=e^{\cos \theta }\cdot e^{-i\sin \theta }----\left(2\right)\end{array}$

On comparing equation $\left(1\right)$ with the equation $\left(2\right)$ we get,

$amplitude\left(r\right)=e^{\cos \theta }$

So, the amplitude of $e^{e^{-i\theta }}$ is equal to $e^{\cos \theta }$.

Hence, option C is the correct answer.