Question

Real part of ${e^e}^{i\theta }$ is

• A. $e^{\cos \left(\theta \right)}\left(\cos \left(\sin \left(\theta \right)\right)\right)$
• B. $e^{\cos \left(\theta \right)}\left(\cos \left(\cos \left(\theta \right)\right)\right)$
• C. $e^{\sin \left(\theta \right)}\left(\sin \left(\cos \left(\theta \right)\right)\right)$
• D. $e^{\sin \left(\theta \right)}\left(\sin \left(\sin \left(\theta \right)\right)\right)$

Answer 1

The correct option is A

$e^{\cos \left(\theta \right)}\left(\cos \left(\sin \left(\theta \right)\right)\right)$

Explanation for the correct option:

Find the Real part of ${e^e}^{i\theta }$:

Given, ${e^e}^{i\theta }$

Then,

$$\begin{gathered} e^{e^{i\theta }}=e^{\cos \theta +i\sin \theta }\left[\because e^{i\theta }=\cos \left(\theta \right)+i\sin \left(\theta \right)\right] \\ =e^{\cos \theta }.e^{i\sin \theta } \\ =e^{\cos \theta }\left(\cos \left(\sin \theta \right)+i\sin \left(\sin \theta \right)\right)\left[\because e^{i\theta }=\cos \left(\theta \right)+i\sin \left(\theta \right)\right] \\ =e^{\cos \theta }.\left(\cos \left(\sin \theta \right)\right)+i{e}^{\cos \theta }.\left(\sin \left(\sin \theta \right)\right) \end{gathered}$$

Therefore, the real part of ${e^e}^{i\theta }$is $e^{\cos \left(\theta \right)}\left(\cos \left(\sin \left(\theta \right)\right)\right)$.

Hence, option (A) is the correct answer.