Question

Rewrite with only sin x and cos x.

sin 3x - cos x

Answer 1

Just to clarify, that's \(\sin3x\), and not \(\sin^3x\), right?

Answer 2

Right.

Answer 3

Okay. Using the angle sum/difference identity, you can write
\[\sin3x=\sin(2x+x)=\sin2x\cos x+\sin x\cos2x\]
You can apply it further to rewrite \(\sin2x\) and \(\cos2x\), or you can use the double angle identities:
\[\sin2x=2\sin x\cos x\\
\cos2x=\cos^2x-\sin^2x\]

Answer 4

My apologizes, but you just lost me.

Answer 5

The first identity I'm referring to is
\[\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\sin\beta\cos\alpha\]
Have you learned this one? It's the only way I can think of to get rid of the \(3x\).

Answer 6

No, I teach myself online. However, I am having difficulties with this section.

Answer 7

In that case, I suggest you hold off on this problem until you become familiar with that identity. While you're at it, you'll probably also learn the angle sum/difference identity for cosine:
\[\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta\]
In any case, I don't think there's another way to do this problem without this identity.

Answer 8

Well, thank you.

Answer 9

You're welcome