Sine rule Questionbank

A student attempts to solve a triangle $LMN$ where side $l = 5$, side $m = 12$, and angle $L = 30^\circ$. How many distinct triangles can be formed with these specific dimensions?

During the conceptual justification of the sine rule, a perpendicular height $h$ is dropped from vertex $A$ to the opposite side $a$. Which pair of expressions correctly represents the height $h$ using right-triangle trigonometry?

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?

Why is the sine rule preferred over the cosine rule when a problem provides two angles and one side ($\text{AAS}$ or $\text{ASA}$)?

In the derivation of the sine rule, if a perpendicular height $h$ is dropped from vertex $C$ to the opposite side $c$, which two expressions for $h$ are equated to link angles $A$ and $B$?

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

When solving a triangle given three side lengths ($SSS$), why can the sine rule NOT be used as the very first step to find an unknown angle?

In any triangle $ABC$, based on the properties of the sine rule, what is the value of the algebraic expression $a \sin C - c \sin A$?