Question

knowledge check - trigonometry
solve for x and y in the triangle below using both law of sines and law of cosines.

Explanation:

Step1: Apply Law of Cosines

By the Law of Cosines, $x^{2}=6.5^{2}+7.1^{2}-2\times6.5\times7.1\times\cos(110^{\circ})$.
First, calculate $\cos(110^{\circ})\approx - 0.342$.
Then $x^{2}=42.25 + 50.41-92.3\times(-0.342)$.
$x^{2}=42.25 + 50.41+31.5666$.
$x^{2}=124.2266$.
$x=\sqrt{124.2266}\approx11.145$.

Step2: Apply Law of Sines

By the Law of Sines, $\frac{\sin(y)}{7.1}=\frac{\sin(110^{\circ})}{x}$.
Since $x\approx11.145$ and $\sin(110^{\circ})\approx0.9397$.
$\sin(y)=\frac{7.1\times0.9397}{11.145}$.
$\sin(y)=\frac{6.67187}{11.145}\approx0.5987$.
$y=\sin^{-1}(0.5987)\approx36.7^{\circ}$.

Answer:

$x\approx11.145$, $y\approx36.7^{\circ}$