Law of Sines Worked Examples Cheat Sheet

Key Facts

• The Law of Sines is $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$, where each side is opposite its matching angle.
• An equivalent form is $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$, which is often convenient when solving for an angle.
• Use $a$ opposite $A$, $b$ opposite $B$, and $c$ opposite $C$ to keep every ratio correctly matched.
• The angles in any triangle satisfy $A+B+C=180^{\circ}$.
• To find a missing side, set up a proportion such as $\frac{a}{\sin A}=\frac{b}{\sin B}$ and solve for the unknown side.
• To find a missing angle, use a proportion such as $\frac{\sin A}{a}=\frac{\sin B}{b}$, then apply $A=\sin^{-1}\left(\frac{a\sin B}{b}\right)$ when solving for $A$.
• The ambiguous case can occur with $SSA$ information because $\sin \theta=\sin(180^{\circ}-\theta)$ may give two possible angles.
• A triangle is impossible if the computed third angle is $0^{\circ}$ or less, or if the side and angle relationships contradict the Law of Sines.

Vocabulary

Law of Sines

A rule that relates each side of a triangle to the sine of its opposite angle using $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$.

Opposite side

The side across from a given angle, such as side $a$ being opposite angle $A$.

Included angle

An angle located between two known sides of a triangle.

SSA case

A triangle information pattern with two sides and a non-included angle, which may produce zero, one, or two possible triangles.

Ambiguous case

A situation in the $SSA$ case where two different triangles can satisfy the same given measurements.

Inverse sine

The operation $\sin^{-1}(x)$ used to find an angle whose sine value is $x$.