## Trigonometry (11th Edition) Clone

A has no value when $0 \lt b \lt 5$
The sum of the three angles in the triangle is $180^{\circ}$. Since $B = 30^{\circ},$ then $0^{\circ} \lt A \lt 150^{\circ}$ We know that $sin~\theta = sin~(180^{\circ}-\theta)$. The angle $A$ has one or two possible values when $0 \lt sin~A \leq 1$ The angle $A$ has no value when $sin~A \gt 1$ We can use the law of sines to find $b$: $\frac{a}{sin~A} = \frac{b}{sin~B}$ $b = \frac{a~sin~B}{sin~A}$ $b = \frac{(10)~sin~30^{\circ}}{sin~A}$ $b = \frac{5}{sin~A}$ If $sin~A \gt 1$ then $0 \lt b \lt 5$ Therefore, A has no value when $0 \lt b \lt 5$