How do you solve the triangle given w=20, x=13, y=12?

Aug 18, 2017

"W," "X," and "Y," refer to the angles opposite sides "w," "x," and "y."

$W = {106.1913516}^{o}$
$X = {38.62483288}^{o}$
$Y = {35.18381552}^{o}$

Explanation:

"W," "X," and "Y," refer to the angles opposite sides "w," "x," and "y."

Start by finding any angle using the Law of Cosines.
$\cos \left(A\right) = \frac{{b}^{2} + {c}^{2} - {a}^{2}}{2 b c}$
Plugging in the numbers, we get:
$\cos \left(W\right) = \frac{{13}^{2} + {12}^{2} - {20}^{2}}{2 \cdot 13 \cdot 12}$
Therefore,
${\cos}^{- 1} \left(\frac{{13}^{2} + {12}^{2} - {20}^{2}}{2 \cdot 13 \cdot 12}\right) = W$
Simplifying inside of the parentheses, we get:
${\cos}^{- 1} \left(\frac{- 29}{104}\right) = W$
Therefore, the measure of angle W is about
${106.1913516}^{o}$

Next, we can apply the Law of Sines.
$\sin \frac{{106.1913516}^{o}}{20} = \sin \frac{X}{13}$
Rearranging the numbers, we get:
$13 \sin \frac{{106.1913516}^{o}}{20} = \sin \left(X\right)$
Therefore,
$X = {\sin}^{- 1} \left(13 \sin \frac{{106.1913516}^{o}}{20}\right)$
Therefore, the measure of angle X is about
${38.62483288}^{o}$

Lastly, we can apply the truth that the sum of all interior angles in a triangle is ${180}^{o}$.
${180}^{o} - {106.1913516}^{o} - {38.62483288}^{o} = Y$
Simplifying, we find the measure of angle Y to be about:
${35.18381552}^{o}$

Therefore, the measure of angle W is ${106.1913516}^{o}$, the measure of angle X is ${38.62483288}^{o}$, and the measure of angle Y is ${35.18381552}^{o}$.