The problem provided in the image can be interpreted as follows:

Given:

• $z = \sqrt{5} (\cos \theta + i \sin \theta)$
• The polar form of $z$ is also given as $z = \sqrt{5} i$

Required:

• Find the value of $\theta$ that satisfies this condition.

Solution:

The polar form of a complex number is expressed as: $z = r (\cos \theta + i \sin \theta)$ where:

• $r$ is the modulus of the complex number (in this case $\sqrt{5}$),
• $\theta$ is the argument (or angle) of the complex number.

From the polar form given as $z = \sqrt{5} i$, we know that the imaginary part is $\sqrt{5}$ and the real part is 0. Thus, the complex number lies on the positive imaginary axis.

This means: $z = 0 + \sqrt{5} i$ We recognize this as a complex number that corresponds to an angle $\theta = \frac{\pi}{2}$ radians (or 90 degrees), because it is purely imaginary and lies on the positive imaginary axis.

Final Answer:

The value of $\theta$ that satisfies the given condition is $\theta = \frac{\pi}{2}$ radians.

Would you like more details or explanations on any part of the solution?

Here are 5 related questions you can explore:

1. What is the general polar form of a complex number?
2. How is the argument $\theta$ determined from a complex number's coordinates?
3. How can the modulus of a complex number be calculated from its Cartesian form?
4. What are other common angles where complex numbers lie on axes (e.g., real or imaginary)?
5. How do you convert between polar and Cartesian forms of a complex number?

Tip: The argument $\theta$ in polar form is always measured from the positive real axis, moving counterclockwise for positive angles.