Question

what is the approximate measure of $\theta$ in radians?
options: $\frac{5\pi}{4}$, $\frac{7\pi}{4}$, $\frac{\pi}{2}$, $\frac{3\pi}{2}$

Explanation:

Step1: Identify full circle radians

A full circle is $2\pi$ radians.

Step2: Analyze shaded portion

The shaded area is $\frac{3}{4}$ of the circle.

Step3: Calculate $\theta$

$\theta = \frac{3}{4} \times 2\pi = \frac{3\pi}{2}$
Verify options: $\frac{3\pi}{2}$ matches the last option, and visually the shaded angle covers 3 quadrants, which is $\frac{3\pi}{2}$ radians.

Answer:

$\boldsymbol{\frac{3\pi}{2}}$ (corresponding to the last option: $\frac{3\pi}{2}$)