To derive the sine rule for angles $A$ and $C$ in triangle $ABC$, an altitude $h$ is dropped from vertex $B$ to side $b$. Which equation correctly expresses $h$ in two different ways using right-triangle trigonometry?

In triangle $ABC$, angle $A=60^{\circ }$, side $a=5$, and side $b=10$. Which of the following statements is correct regarding this triangle?

In triangle $ABC$, you are given side $a=15$, side $b=10$, and angle $A=40^{\circ }$. After calculating the acute value for angle $B\approx 25.4^{\circ }$, a student considers the obtuse candidate $B\approx 154.6^{\circ }$. According to the principle of consistency between sides and angles, why must the obtuse solution be rejected?

In triangle $ABC$, $A=30^{\circ }$, side $b=10$, and side $a=12$. How many distinct triangles can be formed with these dimensions?

A surveyor identifies a triangle $ABC$ where side $AB=100$ m, side $BC=60$ m, and angle $A=30^{\circ }$. How many distinct triangles can be formed with these specific dimensions?