What is argument of complex number e^e^-i theta.i saw this question in SK goyal

We are given the complex number:

e^(e^(-iθ))

To determine its argument, let's break it down step by step.

Step 1: Understanding the Exponential Form
We know that:

e^(-iθ) = cosθ - i sinθ

Thus, our given expression simplifies to:

e^(cosθ - i sinθ)

Step 2: Writing in Polar Form
We express e^(cosθ - i sinθ) as:

e^(cosθ) * e^(-i sinθ)

We can further rewrite this as:

e^(cosθ) * (cos(sinθ) - i sin(sinθ))

Step 3: Identifying the Argument
The argument of a complex number z = r(cosφ + i sinφ) is given by φ.

From our expression:

e^(cosθ) * (cos(sinθ) - i sin(sinθ))

The modulus is e^(cosθ), and the argument is:

sinθ
Final Answer:
The argument of the given complex number is -sinθ.