Question

Law of Sines; B ≈ 42.7°, C ≈ 102.3°, c ≈ 18.7Law of Sines; B ≈ 102.3°, C ≈ 42.7°, c ≈ 18.7Law of Cosines; B ≈ 106.2°, C ≈ 38.8°, c ≈ 18.7Law of Cosines; B ≈ 38.8°, C ≈ 106.2°, c ≈ 18.7

Answer 1

The sine rule is used when we are given either

a) two angles and one side, or

b) two sides and a non-included angle.

The cosine rule is used when we are given either

a) three sides or

b) two sides and the included angle.

For the given problem, we are given a non-included angle and two sides. Hence, we have to solve the problem using the law of sines.

The sine rule states that:

$\frac{\sin\text{ A}}{a}\text{ =}\frac{\sin\text{ B}}{b}\text{ }$

We have:

A = 35 degrees, b = 13, a = 11

Substituting we have:

$\begin{gathered} \frac{\sin35^0}{11}=\text{ }\frac{\sin \text{ B}}{13} \\ \text{Cross}-\text{Multiply} \\ \sin \text{ B }\times11=sin35^0\times13 \end{gathered}$

Divide both sides by 11 and solving for B:

$\begin{gathered} \sin \text{ B = }\frac{\sin \text{ 35 }\times13}{11} \\ \sin \text{ B = 0.677863} \\ B\text{ = 42.68} \\ \approx\text{ 42.7} \end{gathered}$

Using the property of triangles, we can find the angle C:

$\begin{gathered} \angle\text{ A + }\angle\text{ B + }\angle\text{ C =180 (sum of angles in a triangle)} \\ \angle\text{ C = 180 - 42.7 - 35} \\ \angle C\text{=1}02.3 \end{gathered}$

Using the sine rule, we can solve for the unknown side c. We have:

$\begin{gathered} \frac{\sin\text{ C}}{c}=\text{ }\frac{\sin \text{ B}}{b} \\ \frac{\sin\text{ 102.3}}{c}=\text{ }\frac{\sin \text{ 42.7}}{13} \\ \text{Cross}-\text{Multiply} \\ c\text{ }\times\text{ sin 42.7 = sin 102.3 }\times\text{ 13} \\ c\text{ = }\frac{\sin \text{ 102.3 }\times13}{\sin \text{ 42.7}} \\ c\text{ = 18.7295} \\ c\text{ }\approx\text{ 18.7} \end{gathered}$

Answer summary

Law of Sines; B ≈ 42.7°, C ≈ 102.3°, c ≈ 18.7