Question

question 13 (5 points)
determine whether △abc should be solved by using the law of sines or the law of cosines. then solve the triangle.
a = 35°, b = 13, a = 11
law of sines; b ≈ 42.7°, c ≈ 102.3°, c ≈ 18.7
law of sines; b ≈ 102.3°, c ≈ 42.7°, c ≈ 18.7
law of cosines; b ≈ 106.2°, c ≈ 38.8°, c ≈ 18.7
law of cosines; b ≈ 38.8°, c ≈ 106.2°, c ≈ 18.7

Explanation:

Step1: Determine the law to use

Given one - angle and two - sides where the side opposite the given angle and another side are known. So, use the Law of Sines $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$.

Step2: Find angle B

We know that $\frac{a}{\sin A}=\frac{b}{\sin B}$. Substituting $A = 35^{\circ}$, $b = 13$, and $a = 11$, we get $\sin B=\frac{b\sin A}{a}=\frac{13\sin35^{\circ}}{11}$.
$\sin B=\frac{13\times0.5736}{11}=\frac{7.4568}{11}\approx0.6779$.
$B=\sin^{- 1}(0.6779)\approx42.7^{\circ}$ or $B = 180^{\circ}-42.7^{\circ}=137.3^{\circ}$. But since $a < b$, $A=35^{\circ}$, and $A + B<180^{\circ}$, $B\approx42.7^{\circ}$.

Step3: Find angle C

Since the sum of angles in a triangle is $180^{\circ}$, $C=180^{\circ}-A - B$.
$C=180^{\circ}-35^{\circ}-42.7^{\circ}=102.3^{\circ}$.

Step4: Find side c

Using the Law of Sines $\frac{c}{\sin C}=\frac{a}{\sin A}$.
$c=\frac{a\sin C}{\sin A}=\frac{11\times\sin102.3^{\circ}}{\sin35^{\circ}}$.
Since $\sin102.3^{\circ}\approx0.977$, $\sin35^{\circ}\approx0.574$, $c=\frac{11\times0.977}{0.574}=\frac{10.747}{0.574}\approx18.7$.

Answer:

Law of Sines; $B\approx42.7^{\circ}$, $C\approx102.3^{\circ}$, $c\approx18.7$