Question

solve the triangle.
write each answer as an integer or as a decimal rounded to the nearest tenth.
m∠q =
m∠r =
m∠s =

Explanation:

Step1: Use the Law of Cosines to find $\angle Q$

The Law of Cosines formula for finding an angle is $\cos Q=\frac{RS^{2}+QR^{2}-QS^{2}}{2\cdot RS\cdot QR}$. Here, $RS = 3$, $QR=8$, and $QS = 9$. So, $\cos Q=\frac{3^{2}+8^{2}-9^{2}}{2\times3\times8}=\frac{9 + 64-81}{48}=\frac{73 - 81}{48}=\frac{-8}{48}=-\frac{1}{6}$. Then $Q=\cos^{-1}(-\frac{1}{6})\approx 99.6^{\circ}$.

Step2: Use the Law of Sines to find $\angle S$

The Law of Sines formula is $\frac{\sin Q}{RS}=\frac{\sin S}{QR}$. We know $Q\approx99.6^{\circ}$, $RS = 3$, and $QR = 8$. So $\sin S=\frac{QR\cdot\sin Q}{RS}=\frac{8\times\sin(99.6^{\circ})}{3}$. $\sin(99.6^{\circ})\approx0.985$, then $\sin S=\frac{8\times0.985}{3}=\frac{7.88}{3}\approx2.627$ (this is wrong, we should use the Law of Cosines again to find $\angle S$). Using the Law of Cosines: $\cos S=\frac{QS^{2}+RS^{2}-QR^{2}}{2\cdot QS\cdot RS}=\frac{9^{2}+3^{2}-8^{2}}{2\times9\times3}=\frac{81 + 9-64}{54}=\frac{26}{54}\approx0.481$. So $S=\cos^{-1}(0.481)\approx 61.3^{\circ}$.

Step3: Find $\angle R$

Since the sum of the interior angles of a triangle is $180^{\circ}$, $R=180^{\circ}-Q - S$. Substituting $Q\approx99.6^{\circ}$ and $S\approx61.3^{\circ}$, we get $R=180-(99.6 + 61.3)=19.1^{\circ}$.

Answer:

$m\angle Q\approx99.6^{\circ}$
$m\angle R\approx19.1^{\circ}$
$m\angle S\approx61.3^{\circ}$