How to evaluate the limits of $\int (\sin y) dy$? How to evaluate this integral of $$\int^{\arctan2}_{\arctan \frac{1}{2}} \sin y ~dy$$?
Now I know this is $$-[\cos(\arctan 2)-\cos(\arctan 0.5)]=[\cos(\arctan 0.5)-\cos(\arctan 2)]$$ but I have no idea how to figure this out without a calculator.

Look at a $1,2,\sqrt{5}$ triangle and all will reveal itself.
