# A triangle with the hypotenuse measuring 8, and the short leg measuring 5, and the the long leg measuring 7. How do you find the angles C the top angle, B the lower right angle, A the left angle?

Dec 17, 2015

Ignoring the (false) implication that we are dealing with a right angled triangle, we can use the Law of Cosines to determine the 3 angles

#### Explanation:

I have assumed that the triangle looks like:

The reference to a hypotenuse implies a right-angled triangle but based on the Pythagorean Theorem this is clearly false.

The Law of Cosines for a triangle with sides $a , b , c$ and corresponding opposite angles $A , B , C$ is
$\textcolor{w h i t e}{\text{XXX}} {c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cdot \cos \left(C\right)$

which can be re-arranged as:
$\textcolor{w h i t e}{\text{XXX}} \cos \left(C\right) = \frac{\left({a}^{2} + {b}^{2}\right) - {c}^{2}}{2 a b}$
or
$\textcolor{w h i t e}{\text{XXX}} C = \arccos \left(\frac{\left({a}^{2} + {b}^{2}\right) - {c}^{2}}{2 a b}\right)$

and similarly for angles $A$ and $B$

Plugging in the values for $a , b , c$ (I would suggest using a calculator or spreadsheet), we get

$\textcolor{w h i t e}{\text{XXX}} A = 1.047 , B = 1.427 , C = 0.667$

In degrees
$\textcolor{w h i t e}{\text{XXX}} A = {60}^{\circ} , B = {81.8}^{\circ} , C = {38.2}^{\circ}$