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Simplify the expression using a double-angle formula:

[tex]\[ 2 \sin(3\theta) \cos(3\theta) \][/tex]


Sagot :

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]