Question

Write the expression as the sine, cosine, or tangent of an angle.
sin 9x cos x - cos 9x sin x

Answer 1

Recall your Sine Angle Sum Identity:\[\Large\rm \sin \color{orangered}{\alpha}\cos\color{royalblue}{\beta}+\sin\color{royalblue}{\beta} \cos\color{orangered}{\alpha}=\sin(\color{orangered}{\alpha}+\color{royalblue}{\beta})\]

Answer 2

Woops I should have posted the Angle Difference Identity,\[\Large\rm \sin \color{orangered}{\alpha}\cos\color{royalblue}{\beta}-\sin\color{royalblue}{\beta} \cos\color{orangered}{\alpha}=\sin(\color{orangered}{\alpha}-\color{royalblue}{\beta})\]

Answer 3

And we were given:\[\Large\rm \sin \color{orangered}{9x}\cos\color{royalblue}{x}-\sin\color{royalblue}{x} \cos\color{orangered}{9x}\]Understand how to simplify it down using the identity?

Answer 4

Not really

Answer 5

So the formula\[\Large\rm \sin \color{orangered}{\alpha}\cos\color{royalblue}{\beta}-\sin\color{royalblue}{\beta} \cos\color{orangered}{\alpha}=\sin(\color{orangered}{\alpha}-\color{royalblue}{\beta})\]is telling us that when we have this configuration,
we can write it as sine of, and subtract the angles.

So with our problem:\[\Large\rm \sin \color{orangered}{9x}\cos\color{royalblue}{x}-\sin\color{royalblue}{x} \cos\color{orangered}{9x}\]We can apply this formula and write it as:\[\Large\rm \sin(\color{orangered}{9x}-\color{royalblue}{x})\]It can be simplified a tiny bit from there though.

Answer 6

Sin 8x

Answer 7

Yes.