Given $A = 16 ^ { \circ } , b = 12$, and $a = 10$, use the Law of Sines to solve the triangle (if possible) for the value of $c$. If two solutions exist, find both. Round answer to two decimal places.
A) $c = 18.62$
B) $c = 2.10$ and $20.97$
C) $c = 19.75$
D) $c = 1.12$ and $17.85$
E) not possible

Question

Quizplus · Accepted Answer

The Law of Sines states that $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$. Given $A = 16^\circ$, $a = 10$, and $b = 12$, we first find $\sin(A) = \sin(16^\circ)$ and then use the ratio $\frac{\sin(16^\circ)}{10} = \frac{\sin(B)}{12}$ to find $B$. However, since the question asks directly for $c$, we can use the Law of Sines in another form after finding angle $B$ (or directly working with $C$ if $B$ is not needed for the solution). Since the sum of angles in a triangle is $180^\circ$, and we have one angle, we can find another angle and then use the Law of Sines to find $c$. The possibility of two solutions arises from the fact that for an angle $B$, there could be two different angles that satisfy the sine relationship due to the Sine function's properties in different quadrants (acute and obtuse angles). However, without directly calculating $B$ or $C$, we acknowledge the principle that allows for two possible values of $c$ based on the given information, which aligns with choice B, indicating two possible lengths for side $c$, making the triangle solvable with two different configurations.

Given A=16∘,b=12A = 16 ^ { \circ } , b = 12A=16∘,b=12

Given $A = 16 ^ { \circ } , b = 12$