Withough using a calculator, find which of $2\arctan(2\sqrt{2}-1)$ or $3\arctan\left(\frac{1}{4}\right)+\arctan\left(\frac{5}{99}\right)$ is bigger.
I used the formula $\arctan(x)+\arctan(y)=\arctan\left(\frac{x+y}{1-xy}\right)$ to solve this question. Right hand side I found $3\arctan\left(\frac{1}{4}\right)+\arctan\left(\frac{5}{99}\right)$ equals to $\arctan(1)$ and it matches with calculator solution. For left hand side it's not that good. I found $2\arctan(2\sqrt{2}-1)=\arctan\left(-\frac{2+3\sqrt{2}}{4}\right)$ and I said $3\arctan\left(\frac{1}{4}\right)+\arctan\left(\frac{5}{99}\right)$ is the bigger one because of the graph of $\arctan(x)$. But calculator solution is not matches with my solution. Where am I making a mistake. Thanks for all help!
Best Answer
Hint:
Your mistake is hidden in the fact that$$\frac\pi2<\phi\equiv 2\arctan (2\sqrt2-1)<\pi,$$ the negative sign of $\tan(\phi)$ being a clear alarm bell.
If you ignore the bell you will obtain $$\arctan(\tan\phi)=\phi-\pi,$$ since the range of $\arctan x$ is $\left[-\frac\pi2,\frac\pi2\right]$.