Law of Sines
Derive and apply the Law of Sines (a/sinA = b/sinB = c/sinC) to solve non-right triangles given AAS, ASA, or SSA (ambiguous case). Understand the ambiguous case and its implications.
A lesson of Geometry, in Right Triangles and Trigonometry, about 10 minutes.
By the end of this lesson, you will be able to state and apply the Law of Sines to solve oblique (non-right) triangles in AAS and ASA cases, identify the ambiguous SSA case, and find area using the sine formula.
The Big Idea
Right-triangle trig only works for right triangles. The Law of Sines extends our tools to any triangle — acute or obtuse. It's the proportional relationship between each side and the sine of its opposite angle, and it handles any case where you know an angle-side pair.
The Law of Sines
For any triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Equivalently (reciprocal form, sometimes more convenient):
$$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$
When to Use the Law of Sines
| Given information | Case |
|---|---|
| AAS: two angles and a non-included side | ✓ Use Law of Sines |
| ASA: two angles and the included side | ✓ Use Law of Sines |
| SSA: two sides and a non-included angle | ⚠ Ambiguous case |
You always need an angle-opposite-side pair to start.
Example 1 — AAS
$\triangle ABC$: $A = 35°$, $B = 75°$, $a = 10$. Find $b$ and $c$.
First: $C = 180 - 35 - 75 = 70°$
$$\frac{10}{\sin 35°} = \frac{b}{\sin 75°}$$
$$b = \frac{10\sin 75°}{\sin 35°} \approx \frac{10(0.9659)}{0.5736} \approx 16.84$$
$$\frac{10}{\sin 35°} = \frac{c}{\sin 70°}$$
$$c = \frac{10\sin 70°}{\sin 35°} \approx \frac{10(0.9397)}{0.5736} \approx 16.38$$
Example 2 — ASA
$\triangle ABC$: $A = 50°$, $C = 65°$, $b = 20$. Find $a$ and $c$.
$B = 180 - 50 - 65 = 65°$. Since $B = C = 65°$, this is isosceles: $b = c = 20$.
For $a$: $\dfrac{20}{\sin 65°} = \dfrac{a}{\sin 50°}$
$a = \dfrac{20\sin 50°}{\sin 65°} \approx \dfrac{20(0.766)}{0.906} \approx 16.91$
The Ambiguous Case (SSA)
When given two sides and an angle opposite one of them (SSA), there may be:
- No triangle (the side opposite the angle is too short to reach)
- One triangle (exactly tangent or angle ≥ 90°)
- Two triangles (two different triangles satisfy the conditions)
Checking SSA: Given $a$, $b$, and angle $A$ (opposite $a$):