# How do you solve the triangle when given a=5, b=7, and c=6?

Nov 28, 2015

Use the Law of Cosines to determine the angles

#### Explanation:

Law of Cosines
$\textcolor{w h i t e}{\text{XXX}} {c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos \left(\angle C\right)$
$\Rightarrow$
$\textcolor{w h i t e}{\text{XXX}} \cos \left(C\right) = \frac{{a}^{2} + {b}^{2} - {c}^{2}}{2 a b}$

Similarly
$\textcolor{w h i t e}{\text{XXX}} \cos \left(A\right) = \frac{{b}^{2} + {c}^{2} - {a}^{2}}{2 b c}$
and
$\textcolor{w h i t e}{\text{XXX}} \cos \left(B\right) = \frac{{a}^{2} + {c}^{2} - {b}^{2}}{2 a c}$

Substituting the given values for $a , b , \mathmr{and} c$
we get
$\textcolor{w h i t e}{\text{XXX}} \cos \left(\angle A\right) = 0.714$
then taking the $\arccos$ of both sides:
color(white)("XXX")/_A = arcos(0.715) = 0.775 " radians" = 44.4^@

Similarly we can find:
color(white)("XXX")/_B = 1.369 " radians" = 78.5^@
and
color(white)("XXX")/_C = 0.997 " radians" = 57.1^@