How can I find radians without a calculator's trigonometry functions?

[The Student]

Using c (hypotenuse) = 1, x = 0.8 (horizontal), how can I know the radians without any use of a calculator?

 

[MarkFL]

Form the given information we know:

$\displaystyle \cos(\theta)=\dfrac{0.8}{1}=\dfrac{4}{5}$ and so:

$\displaystyle \theta=\cos^{-1}\left(\dfrac{4}{5} \right)$

This does not correspond to a special angle, so without a calculator or table one option would be to approximate the angle by truncating the following Maclaurin series:

$\displaystyle \cos^{-1}(x)=\dfrac{\pi}{2}-\sum_{k=0}^{\infty}\dfrac{(2k)!}{4^k(k!)^2(2k+1)}x^{2k+1}$ where $\displaystyle |x|\le1$

 

[The Student]

Form the given information we know:

$\displaystyle \cos(\theta)=\dfrac{0.8}{1}=\dfrac{4}{5}$ and so:

$\displaystyle \theta=\cos^{-1}\left(\dfrac{4}{5} \right)$

This does not correspond to a special angle, so without a calculator or table one option would be to approximate the angle by truncating the following Maclaurin series:

$\displaystyle \cos^{-1}(x)=\dfrac{\pi}{2}-\sum_{k=0}^{\infty}\dfrac{(2k)!}{4^k(k!)^2(2k+1)}x^{2k+1}$ where $\displaystyle |x|\le1$

Oh wow, thanks a lot. I expected it to be much easier, so now I know why we just learnt the special angles in grade 12.