# How do you use the law of cosines or law of sines if you are given a = 55, b = 25, c = 72?

May 22, 2015

The law of cosine is:

${a}^{2} = {b}^{2} + {c}^{2} - 2 b c \cos \alpha$,

where $\alpha$ is the opposite angle of $a$ (and so on for every angle of the triangle).

So:

$\cos \alpha = \frac{{b}^{2} + {c}^{2} - {a}^{2}}{2 b c} = \frac{{25}^{2} + {72}^{2} - {55}^{2}}{2 \cdot 25 \cdot 72} = 0.773$

$\cos \beta = \frac{{a}^{2} + {c}^{2} - {b}^{2}}{2 a c} = \frac{{55}^{2} + {72}^{2} - {25}^{2}}{2 \cdot 55 \cdot 72} = 0.957$

$\cos \gamma = \frac{{a}^{2} + {b}^{2} - {c}^{2}}{2 a b} = \frac{{55}^{2} + {25}^{2} - {72}^{2}}{2 \cdot 55 \cdot 25} = - 0.558$.

So:

alpha~=39.37°

beta~=16.86°

gamma~=123.92°,

using the inverse function of the function $y = \cos x$, that is $y = \arccos x$.