Oblique Triangles

Overview

An oblique triangle is any triangle that is NOT a right triangle. We use the Law of Sines and Law of Cosines to solve these triangles.

Standard Notation

• Angles: $A$, $B$, $C$ (capital letters)
• Opposite sides: $a$, $b$, $c$ (lowercase letters)
• Side $a$ is opposite angle $A$, etc.

Law of Sines

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Or equivalently:

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

When to Use Law of Sines

• AAS: Two angles and a non-included side
• ASA: Two angles and the included side
• SSA: Two sides and an angle opposite one of them (ambiguous case)

Law of Cosines

$a^2 = b^2 + c^2 - 2bc \cos A$

$b^2 = a^2 + c^2 - 2ac \cos B$

$c^2 = a^2 + b^2 - 2ab \cos C$

Solving for Angles

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

$\cos B = \frac{a^2 + c^2 - b^2}{2ac}$

$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$

When to Use Law of Cosines

• SAS: Two sides and the included angle
• SSS: All three sides

The Ambiguous Case (SSA)

When given two sides and an angle opposite one of them, there may be:

• No solution
• One solution
• Two solutions

Given: sides $a$, $b$ and angle $A$ (angle opposite side $a$)

Let $h = b \sin A$ (the height)

Condition	Number of Solutions
$a < h$	0 (no triangle)
$a = h$	1 (right triangle)
$h < a < b$	2 (ambiguous)
$a \geq b$	1

Solving the Ambiguous Case

1. Use Law of Sines to find $\sin B$
2. If $\sin B > 1$: No solution
3. If $\sin B \leq 1$: $B = \arcsin(\sin B)$
4. Check if $B' = 180° - B$ also works
5. Verify that $A + B < 180°$ (and $A + B' < 180°$)

Area Formulas

Using Two Sides and Included Angle

$\text{Area} = \frac{1}{2}ab \sin C$

$\text{Area} = \frac{1}{2}bc \sin A$

$\text{Area} = \frac{1}{2}ac \sin B$

Using All Three Sides (Heron's Formula)

$s = \frac{a + b + c}{2} \quad \text{(semi-perimeter)}$

$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$

Solving Strategy

Given	Use
AAA	Cannot solve (need at least one side)
AAS/ASA	Law of Sines
SSA	Law of Sines (check ambiguous case)
SAS	Law of Cosines, then Law of Sines
SSS	Law of Cosines, then Law of Sines

Step-by-Step Examples

SAS Example

Given: $a = 8$, $b = 6$, $C = 60°$

1. Find $c$ using Law of Cosines:

$c^2 = a^2 + b^2 - 2ab \cos C$

$c^2 = 64 + 36 - 2(8)(6)(0.5)$

$c^2 = 100 - 48 = 52$

$c = \sqrt{52} \approx 7.21$

2. Find angle $A$ using Law of Sines:

$\frac{\sin A}{a} = \frac{\sin C}{c}$

$\sin A = \frac{a \sin C}{c} = \frac{8(0.866)}{7.21} \approx 0.961$

$A \approx 73.9°$

3. Find angle $B$:

$B = 180° - A - C = 180° - 73.9° - 60° \approx 46.1°$

Key Relationships

• Sum of angles: $A + B + C = 180°$
• Triangle inequality: Each side $<$ sum of other two
• Largest angle opposite largest side
• Law of Cosines reduces to Pythagorean theorem when angle $= 90°$