To express \(\sin(3x)\) in terms of \(\sin(x)\) and \(\cos(x)\), you can use trigonometric identities. The triple-angle identity for sine is particularly useful:

\[
\sin(3x) = 3\sin(x) - 4\sin^3(x)
\]

To change it into a form that involves \(\sin(x)\) and \(\cos(x)\), use the identity:

\[
\sin^2(x) = 1 - \cos^2(x)
\]

Now, substitute \(\sin^3(x)\) with \(\sin(x)(1 - \cos^2(x))\):

\[
\sin(3x) = 3\sin(x) - 4\sin(x)(1 - \cos^2(x))
\]

Simplify the expression:

\[
\sin(3x) = 3\sin(x) - 4\sin(x) + 4\sin(x)\cos^2(x)
\]
\[
\sin(3x) = -\sin(x) + 4\sin(x)\cos^2(x)
\]
\[
\sin(3x) = \sin(x)(4\cos^2(x) - 1)
\]

So, \(\sin(3x)\) can be rewritten in the form:

\[
\sin(3x) = \sin(x)(4\cos^2(x) - 1)
\]

This expression still involves \(\sin(x)\) and \(\cos(x)\) but in a combined form.

### Would you like further details or have any questions?

Here are some related questions:
1. What is the double-angle identity for sine?
2. How do you express \(\cos(3x)\) using \(\cos(x)\) and \(\sin(x)\)?
3. What is the triple-angle identity for cosine?
4. How do you derive the triple-angle identity for sine?
5. Can \(\sin(4x)\) be expressed using \(\sin(x)\) and \(\cos(x)\)?
6. How do you expand \(\sin(2x)\) using trigonometric identities?
7. What is the sum-to-product identity for sine and cosine?
8. How do you express higher powers of sine or cosine using trigonometric identities?

**Tip:** Familiarizing yourself with trigonometric identities, like the angle-sum and multiple-angle formulas, can significantly simplify complex expressions.

Discover how to convert sin(3x) to sin(x) cos(x) using trigonometric identities. This detailed explanation covers the triple-angle identity for sine and the sine squared identity, providing step-by-step solutions. Suitable for advanced high school students and anyone studying trigonometry.

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