Trigonometry (10th Edition)

A has exactly one value when $b \geq 10$ or $b = 5$
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 $sin~\theta = sin~(180^{\circ}-\theta)$. Therefore, the angle $A$ has exactly one value when $0 \lt A \leq 30^{\circ}$ or $A = 90^{\circ}$ Then $0 \lt sin~A \leq 0.5$ or $sin~A = 1$ 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 $0 \lt sin~A \leq 0.5$ then $b \geq 10$ If $sin~A = 1$ then $b = 5$ Therefore, A has exactly one value when $b \geq 10$ or $b = 5$