a.

The sine law should be used. The 34° angle and 42 m form a matching pair. The angle given, 34°, is not between the two known side lengths. The side length opposite the unknown angle is also given.

b.

Label ΔABC.


The given information in ΔABC is

$\begin{array}{rcl}\angle A& =& 34°\\ a& =& 42 \mathrm{m}\\ b& =& 60 \mathrm{m}\end{array}$

$\angle A=34°$ is across from side a = 42 m. ($\angle A$ and side a are a matching pair.)
$\angle B$ is across from side b = 60 m. ($\angle B$ and side b are a matching pair.)

To find the unknown angle, use the formula $\frac{\sin A}{a}=\frac{\sin {\displaystyle B}}{b}$.

$\begin{array}{rcl}\frac{\sin A}{a}& =& \frac{\sin {\displaystyle B}}{b}\\ \frac{\sin {\displaystyle B}}{b}& =& \frac{\sin {\displaystyle A}}{a}\\ \frac{\sin {\displaystyle B}}{60}& =& \frac{\sin {\displaystyle 34}{\displaystyle °}}{42}\\ \frac{\sin {\displaystyle B}}{\cancel{60}}\times \cancel{60}& =& \frac{\sin {\displaystyle 34}{\displaystyle °}}{42}\times 60\\ \sin B& =& \frac{\sin {\displaystyle 34}{\displaystyle °}}{42}\times 60\\ & =& 0.7988\\ \angle B& =& {\sin }^{-1}0.7988\\ & =& 53°\\ & & \end{array}$
Since $\angle B=\angle R, \angle R$ is equal to 53°.

a.

The sine law should be used. The 28° angle and 42 km form a matching pair. The angle given, 106°, is not between the two known side lengths.

b.

Label ΔABC.


The given information in ΔABC is

$\begin{array}{rcl}\angle A& =& 106°\\ \angle B& =& 28°\\ b& =& 42 \mathrm{km}\end{array}$

To find a side, use the formula $\frac{a}{\sin A}=\frac{b}{\sin {\displaystyle B}}$.

$\begin{array}{rcl}\frac{a}{\sin A}& =& \frac{b}{\sin {\displaystyle B}}\\ \frac{a}{\sin {\displaystyle 106}{\displaystyle °}}& =& \frac{42}{\sin {\displaystyle 28}{\displaystyle °}}\\ \frac{a}{{\displaystyle \cancel{\sin 106°}}}\times \cancel{\sin 106°}& =& \frac{42}{\sin {\displaystyle 28}{\displaystyle °}}\times \sin 106°\\ a& =& \frac{42}{\sin {\displaystyle 28}{\displaystyle °}}\times \sin 106°\\ & =& 86.0\\ & & \end{array}$

The distance from ship B to ship C is 86.0 km.

a.

The cosine law should be used since all three side lengths are known and no angles are given.

b.

Label ΔABC.


The given information in ΔABC is

a = 70 cm
b = 120 cm
c = 100 cm

Substitute the known sides into the cosine law to solve for $\angle A$.

$\begin{array}{rcl}\cos A& =& \frac{b^2+c^2-a^2}{2bc}\\ \cos A& =& \frac{{120}^2+{100}^2-{70}^2}{2\left(120\right)\left(100\right)}\\ & =& \frac{14 400+10 000-4 900}{24 000}\\ & =& \frac{19 500}{24 000}\\ & =& 0.8125\\ \angle A& =& {\cos }^{-1}0.8125\\ & =& 36°\end{array}$

$\angle A$ has an angle measure of 36°.