# Calculate sum of angles if you know their tan value

$$\tan(u) = 2, \ \ \ 0 < u < \frac{\pi}{2}\\ \tan(v) = 3, \ \ \ 0 < v < \frac{\pi}{2}$$

What is $u + v$?

I know that both angles are in the first quadrant in the unit circle. How do I calculate the sum of the angles? (Without calculator of course.) I'm not sure on how to think, or what methods to apply. I know that $\tan(u) = \frac{\sin(u)}{\cos(u)}$, but I'm not sure that helps here.

• If you know what $\arctan$ is, the problem is almost solved, and the answer is $u+v=\arctan 2 + \arctan 3$.
– Lehs
Nov 27 '14 at 10:14
• @Lehs Won't I need a calculator for that? Nov 27 '14 at 10:15

Use the formula $$\tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x\tan y}.$$
$$\tan(u+v)=-1\implies u+v=n\pi-\frac\pi4$$ where $n$ is any integer
As $0<u+v<\pi, n=1$