# What is cos^-1(5/13)?

Aug 18, 2015

${\cos}^{- 1} \left(\frac{5}{13}\right) = 1.176005 + n \pi , \forall n \in \mathbb{Z}$

#### Explanation:

${\cos}^{- 1} \left(\frac{5}{13}\right) = \theta$
$\Rightarrow$$\textcolor{w h i t e}{\text{XXXX}}$$\cos \left(\theta\right) = \frac{5}{13}$

This ratio gives us a Pythagorean triangle
with hypotenuse $= 13$
$\textcolor{w h i t e}{\text{XX}}$adjacent side $= 5$
$\textcolor{w h i t e}{\text{XX}}$opposite side $= 12$

Unfortunately this is not one of the standard triangles with a simple angle.
The only ways to solve it are by using a calculator, or trig tables, or an infinite series approximation.
This gives $\theta = 1.176005$ radians

Note that the same $5 : 13$ triangular ratios also exist with a congruent reference triangle in Quadrant III.
Therefore adding $\pi$ (and any integer multiple of $\pi$) will also give a valid value for $\theta$.