Certainly! Let's solve this step by step.
We are given that [tex]\(\sin \theta = \frac{3}{5}\)[/tex]. We need to find the value of [tex]\(\sin 2\theta\)[/tex].
1. Find [tex]\(\cos \theta\)[/tex]:
Using the Pythagorean identity:
[tex]\[\sin^2 \theta + \cos^2 \theta = 1\][/tex]
We substitute [tex]\(\sin \theta = \frac{3}{5}\)[/tex] into the equation:
[tex]\[\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1\][/tex]
[tex]\[\frac{9}{25} + \cos^2 \theta = 1\][/tex]
[tex]\[\cos^2 \theta = 1 - \frac{9}{25}\][/tex]
[tex]\[\cos^2 \theta = \frac{25}{25} - \frac{9}{25}\][/tex]
[tex]\[\cos^2 \theta = \frac{16}{25}\][/tex]
Taking the positive square root (assuming [tex]\(\theta\)[/tex] is in a quadrant where cosine is positive), we get:
[tex]\[\cos \theta = \frac{4}{5}\][/tex]
2. Use the double-angle formula for sine:
The double-angle formula for sine is:
[tex]\[\sin 2\theta = 2 \sin \theta \cos \theta\][/tex]
3. Calculate [tex]\(\sin 2\theta\)[/tex]:
Substitute [tex]\(\sin \theta = \frac{3}{5}\)[/tex] and [tex]\(\cos \theta = \frac{4}{5}\)[/tex] into the formula:
[tex]\[\sin 2\theta = 2 \times \frac{3}{5} \times \frac{4}{5}\][/tex]
[tex]\[\sin 2\theta = 2 \times \frac{12}{25}\][/tex]
[tex]\[\sin 2\theta = \frac{24}{25}\][/tex]
Thus, the value of [tex]\(\sin 2\theta\)[/tex] is [tex]\(\frac{24}{25}\)[/tex] or equitively [tex]\(0.96\)[/tex].
In the process, we also found [tex]\(\cos \theta = \frac{4}{5}\)[/tex] or [tex]\(0.8\)[/tex].
