# How do you solve the triangle given a=21.5, b=13, C=38?

Feb 17, 2018

$c = 13.8113$, $B = {35.4151}^{o}$, $A = {106.5849}^{o}$

#### Explanation:

We use law of cosines to find side c.
${c}^{2} = {\left(21.5\right)}^{2} + {\left(13\right)}^{2} - 2 \left(21.5\right) \left(13\right) \cos \left(38\right)$.
Square rooting both sides, we get $c = 13.8113$.
From there, we can use law of sines to find angle B.
$\sin \frac{B}{13} = \sin \frac{38}{9.8673}$.
Solving, we get $\sin \left(B\right) = \frac{13 \sin \left(38\right)}{13.8113}$.
Using ${\sin}^{-} 1$, we get ${\sin}^{-} 1 \left(\frac{13 \sin \left(38\right)}{13.8113}\right) = {35.4151}^{o}$
Subtracting from 180 to find the third angle,we get
$A = 180 - 38 - 35.4151 = {106.5849}^{o}$