TITLE: How to find the inverse cosine without a calculator
QUESTION [11 upvotes]: How to find the inverse of:
$$\cos(c)=\frac{1}{3}$$
In other words, i'm trying to solve for c and without a calculator. If it's hard or not possible, then how would you go about solving inverses in general. For example, lets say:
$$\cos(c)=\frac{3+1}{2\sqrt{2}}$$
how would we solve for c, which in this case is 105

REPLY [2 votes]: $$\arccos x =\dfrac{\pi}{2}-\displaystyle\sum_{n=0}^{\infty} \dfrac{(2n)!}{2^{2n}(n!)^2(2n+1)}x^{2n+1}=\dfrac{\pi}{2}-\biggl(x+\dfrac{1}{2}\dfrac{x^3}{3}+\dfrac{1}{2\cdot4}\dfrac{x^5}{5}+\underbrace{\cdots}_{o(x)}\biggr), \mid x\mid \lt 1$$

$$\arccos\biggl(\dfrac{1}{3}\biggr)\approx\dfrac{\pi}{2}-\biggl(\dfrac{1}{3}+\dfrac{1}{162}+\dfrac{1}{9720}+o\biggl(\dfrac{1}{3}\biggr)\biggr)\approx 1.231290=70.54773^{\circ}$$