To determine the measure of the central angle in radians for a sector where the ratio of the area of the sector to the area of the circle is [tex]\(\frac{3}{5}\)[/tex], follow these steps:
1. Understanding the Relationship: The ratio of the area of a sector to the total area of the circle is equal to the ratio of the central angle of the sector to the full angle of the circle (which is [tex]\(2\pi\)[/tex] radians).
[tex]\[
\text{Ratio of sector area to total area} = \frac{\text{Central angle}}{2\pi}
\][/tex]
2. Given Ratio: We are given that the ratio of the area of sector [tex]\(AOB\)[/tex] to the area of the circle is [tex]\(\frac{3}{5}\)[/tex].
[tex]\[
\frac{\theta}{2\pi} = \frac{3}{5}
\][/tex]
3. Solving for [tex]\(\theta\)[/tex]:
Multiply both sides of the equation by [tex]\(2\pi\)[/tex] to isolate [tex]\(\theta\)[/tex]:
[tex]\[
\theta = 2\pi \times \frac{3}{5}
\][/tex]
4. Calculate [tex]\(\theta\)[/tex]:
Substituting the value of [tex]\(\pi \approx 3.14159\)[/tex],
[tex]\[
\theta = 2 \times 3.14159 \times \frac{3}{5} = 6.28318 \times \frac{3}{5} = 6.28318 \times 0.6 = 3.76991
\][/tex]
5. Rounding:
Can round the result to two decimal places:
[tex]\[
\theta \approx 3.77
\][/tex]
6. Comparing with Options:
The approximate measure of the central angle corresponding to [tex]\(\widehat{AB}\)[/tex] in radians is [tex]\(3.77\)[/tex].
7. Conclusion:
Therefore, the correct choice is [tex]\( \boxed{3.77} \)[/tex].
