Question

State whether the given measurements determine zero, one, or two triangles. b=86, b=21, c=18

Answer 1

Answer: 1

Step-by-step explanation:

I assume you meant to type $B=86^{\circ}, b=21, c=18$.

By the Law of Sines,

$\frac{\sin C}{c}=\frac{\sin B}{b} \\ \\ \frac{\sin C}{18}=\frac{\sin 86^{\circ}}{21} \\ \\ \sin C=\frac{18 \sin 86^{\circ}}{21} \\ \\ C=\sin^{-1} \left(\frac{18 \sin 86^{\circ}}{21} \right), 180^{\circ}-\sin^{-1} \left(\frac{18 \sin 86^{\circ}}{21} \right) \\ \\ C \approx 58.77^{\circ}, 121.23^{\circ}$

Since only one of these values will make a triangle (the obtuse possibility for C will mean $B+C > 180^{\circ}$, which violates the condition that the angles of a triangle add to 180 degrees), 1 triangle can be formed.