Sure! Let's simplify the expression [tex]\( 2 \sin 3\theta \cos 3\theta \)[/tex] by using a double-angle formula.
First, recall the double-angle formula for sine:
[tex]\[
\sin 2x = 2 \sin x \cos x
\][/tex]
To use this formula for our given expression, we need to recognize how we can match the form of [tex]\( 2 \sin x \cos x \)[/tex]. Notice that in our case:
[tex]\[
2 \sin 3\theta \cos 3\theta
\][/tex]
Here, [tex]\( x \)[/tex] is substituted by [tex]\( 3\theta \)[/tex]. So, setting [tex]\( x = 3\theta \)[/tex], we get:
[tex]\[
2 \sin 3\theta \cos 3\theta = \sin(2 \cdot 3\theta)
\][/tex]
Simplifying further:
[tex]\[
\sin(2 \cdot 3\theta) = \sin 6\theta
\][/tex]
Thus, the expression [tex]\( 2 \sin 3\theta \cos 3\theta \)[/tex] simplifies to:
[tex]\[
\sin 6\theta
\][/tex]
Therefore, the simplified form of [tex]\( 2 \sin 3\theta \cos 3\theta \)[/tex] is:
[tex]\[
\sin 6\theta
\][/tex]
