To solve for the measure of [tex]\(\theta\)[/tex] in radians given the radius [tex]\(r = 3\)[/tex] units and the area of the sector [tex]\(\frac{3\pi}{2}\)[/tex] square units, follow these steps:
1. Recall the formula for the area of a sector:
The area [tex]\(A\)[/tex] of a sector of a circle with radius [tex]\(r\)[/tex] and angle [tex]\(\theta\)[/tex] (in radians) is given by:
[tex]\[
A = \frac{1}{2} r^2 \theta
\][/tex]
2. Substitute the given values into the formula:
[tex]\[
\frac{3\pi}{2} = \frac{1}{2} \times 3^2 \times \theta
\][/tex]
Simplify the radius squared:
[tex]\[
\frac{3\pi}{2} = \frac{1}{2} \times 9 \times \theta
\][/tex]
3. Solve for [tex]\(\theta\)[/tex]:
[tex]\[
\frac{3\pi}{2} = \frac{9}{2} \theta
\][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[
3\pi = 9\theta
\][/tex]
Divide both sides by 9:
[tex]\[
\theta = \frac{3\pi}{9}
\][/tex]
Simplify the fraction:
[tex]\[
\theta = \frac{\pi}{3}
\][/tex]
4. Express [tex]\(\theta\)[/tex] as [tex]\(\frac{\pi}{n}\)[/tex]:
From the previous step, we have [tex]\(\theta = \frac{\pi}{3}\)[/tex], which means [tex]\(n = 3\)[/tex].
So, the measure of [tex]\(\theta\)[/tex] in radians is [tex]\(\frac{\pi}{3}\)[/tex] radians. Hence, [tex]\(n = 3\)[/tex].
