To solve the problem of finding [tex]\(\sin 2\theta\)[/tex] given [tex]\(\cos \theta = \frac{4}{5}\)[/tex] and [tex]\(0^\circ < \theta < 90^\circ\)[/tex], we will follow these steps:
1. Calculate [tex]\(\sin \theta\)[/tex]:
Given the Pythagorean identity:
[tex]\[
\cos^2 \theta + \sin^2 \theta = 1
\][/tex]
We can solve for [tex]\(\sin \theta\)[/tex]:
[tex]\[
\sin^2 \theta = 1 - \cos^2 \theta
\][/tex]
Substituting [tex]\(\cos \theta = \frac{4}{5}\)[/tex]:
[tex]\[
\cos^2 \theta = \left(\frac{4}{5}\right)^2 = \frac{16}{25}
\][/tex]
Thus,
[tex]\[
\sin^2 \theta = 1 - \frac{16}{25} = \frac{9}{25}
\][/tex]
Taking the positive square root (since [tex]\(0^\circ < \theta < 90^\circ\)[/tex] and sine is positive in this interval):
[tex]\[
\sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5}
\][/tex]
2. Calculate [tex]\(\sin 2\theta\)[/tex]:
Using the double angle formula for sine:
[tex]\[
\sin 2\theta = 2 \sin \theta \cos \theta
\][/tex]
Substituting the values [tex]\(\sin \theta = \frac{3}{5}\)[/tex] and [tex]\(\cos \theta = \frac{4}{5}\)[/tex]:
[tex]\[
\sin 2\theta = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = 2 \cdot \frac{12}{25} = \frac{24}{25}
\][/tex]
Therefore, [tex]\(\sin 2\theta = \frac{24}{25}\)[/tex].
The correct answer is [tex]\( \text{b.} \frac{24}{25} \)[/tex].
So, the best answer from the choices provided is:
B
