What is this type of problem called?

Question

I've looked in my book and all over the internet trying to make sense of problems like this:

$\sin(\arctan(\frac{3}{4}))$

What is this type of problem called? I don't know what to search for.

Answer 1

The name for this kind of problem is a trigonometric function using the inverse of $\tan$ in the form of $\arctan$.

Answer 2

"involving trig and inverse trig functions" generates a few hits.

what you need to do to solve problems like these is to translate the trig function into a form that cancels easily with the inverse trig function. For you example, you need to express the $\sin$ in terms of $\tan$s so that you get a bunch of $\tan(arctan(3/4))$ in your expression.

For your example, here's what I would do $$\tan x = \frac{\sin x}{\cos x}$$ $$\tan x = \sin x\sec x$$ $$\tan x = \sin x\sqrt{\tan^2 x+1}$$ $$\sin x = \frac{\tan x}{\sqrt{\tan^2 x+1}}$$ Now we have an identity, we apply it to the problem $$\sin(\arctan 3/4)=\frac{\tan (\arctan 3/4)}{\sqrt{\tan^2 (\arctan 3/4)+1}}=\frac{3/4}{\sqrt{(3/4)^2+1}}=3/5$$