Given that \(\sin \theta = \frac{9}{41}\) and \(\theta\) is in the first quadrant, we need to find the exact value of \(\sin 2\theta\).
1. Recollect the identity for \(\sin 2\theta\):
[tex]\[
\sin 2\theta = 2 \sin \theta \cos \theta
\][/tex]
2. Find \(\cos \theta\) using the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Given \(\sin \theta = \frac{9}{41}\), we substitute and solve for \(\cos \theta\):
[tex]\[
\left(\frac{9}{41}\right)^2 + \cos^2 \theta = 1
\][/tex]
[tex]\[
\left(\frac{81}{1681}\right) + \cos^2 \theta = 1
\][/tex]
[tex]\[
\cos^2 \theta = 1 - \frac{81}{1681}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{1681}{1681} - \frac{81}{1681}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{1600}{1681}
\][/tex]
Since \(\theta\) is in the first quadrant where cosine is positive:
[tex]\[
\cos \theta = \sqrt{\frac{1600}{1681}} = \frac{40}{41}
\][/tex]
3. Apply the double-angle formula using \(\sin \theta = \frac{9}{41}\) and \(\cos \theta = \frac{40}{41}\):
[tex]\[
\sin 2\theta = 2 \sin \theta \cos \theta
\][/tex]
[tex]\[
\sin 2\theta = 2 \left(\frac{9}{41}\right) \left(\frac{40}{41}\right)
\][/tex]
[tex]\[
\sin 2\theta = 2 \left(\frac{360}{1681}\right)
\][/tex]
[tex]\[
\sin 2\theta = \frac{720}{1681}
\][/tex]
Thus, the exact value of [tex]\(\sin 2\theta\)[/tex] is [tex]\(\frac{720}{1681}\)[/tex].
