Question

In many cases the Law of Sines works perfectly well and returns the correct missing values in a non-right triangle. However, in some cases the Law of Sines returns two possible measurements.Consider the diagram below, and assume that m∠A=29∘, AC=13 cm, and, BC=7.2 cm.Using the Law of Sines, determine the value of m∠B. You should notice that there are actually two possible values - list both of them (separated by a comma).m∠B=  °   If we assume the diagram is to scale, which value of m∠B makes more sense? Enter the appropriate value.m∠B= °   Using your answer to part (b), determine the length of AB.¯¯¯¯¯¯¯¯AB=  cm   

Answer 1

$\frac{\sin B}{b}=\frac{\sin A}{a} \\ \\ \sin B=\frac{b \sin A}{a} \\ \\ \sin B=\frac{13 \sin 29^{\circ}}{7.2} \\ \\ \boxed{m\angle B =\arcsin \left(\frac{65\sin 29^{\circ}{36} \right), 180^{\circ}-\arcsin \left(\frac{65\sin 29^{\circ}}{36} \right)}$

Considering the diagram. $\angle B$ is shown to be acute, so $\boxed{m\angle B=\arcsin \left(\frac{65\sin 29^{\circ}}{36} \right)}$.

Angles in a triangle add to 180°, so $m\angle C= 151^{\circ}-\arcsin \left(\frac{65\sin 29^{\circ}}{36} \right)$.

Using the law of cosines,

$AB=\sqrt{13^2 + 7.2^2-2(13)(7.2)\cos \left(\arcsin \left(\frac{65\sin 29^{\circ}}{36} \right) \right)} \\ \\ \boxed{AB=\sqrt{220.84-187.2\cos \left(\arcsin \left(\frac{65\sin 29^{\circ}}{36} \right) \right)}}$