Triangle Geometry and the Law of Sines

Triangle Geometry and Trigonometry

Sum of Interior Angles in a Triangle

In Euclidean geometry, the sum of the interior angles of any triangle is always 180°. This fundamental property is essential for solving many problems involving triangles.

• Key Point: For any triangle with angles A, B, and C:

$A + B + C = 180^{\circ}$

• Example: If two angles of a triangle are 50° and 60°, the third angle is $180^{\circ} - 50^{\circ} - 60^{\circ} = 70^{\circ}$.

Right Triangle Trigonometry

Right triangle trigonometry is used to solve problems involving right triangles, where one angle is exactly 90°. The relationships between the sides and angles are described by trigonometric ratios: sine, cosine, and tangent.

• Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• Calculator Mode: When solving for angles or sides, ensure your calculator is in degree mode if angles are given in degrees.

• Example: Given a right triangle with one leg of 8.39 ft, another leg of 12.44 ft, and an angle of 34°, use the above ratios to find missing sides or angles.

Solving Triangles: Cases and Methods

Triangle Solving Cases

To solve a triangle means to find all its side lengths and angle measures. Depending on the information given, different cases arise:

• Angle-Angle-Side (AAS): Two angles and a non-included side are known.

• Angle-Side-Angle (ASA): Two angles and the included side are known.

• Side-Side-Angle (SSA): Two sides and a non-included angle are known. This is also called the "ambiguous case" because it may yield one, two, or no solutions.

Note: When solving for angles in the SSA case, be aware of the possibility of multiple solutions (ambiguous case).

Law of Sines

Definition and Formula

The Law of Sines relates the sides and angles of any triangle (not just right triangles). It is especially useful for solving triangles in the AAS, ASA, and SSA cases.

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

• A, B, C: Angles of the triangle

• a, b, c: Sides opposite angles A, B, and C, respectively

Applications and Examples

• Solving for a Side: If two angles and one side are known, use the Law of Sines to find an unknown side.

• Solving for an Angle: If two sides and a non-included angle are known, use the Law of Sines to find an unknown angle.

• Example 1 (AAS/ASA): Given $A = 20^{\circ}$, $B = 114^{\circ}$, $a = 65$, find $b$ and $c$:

  • First, find $C = 180^{\circ} - 20^{\circ} - 114^{\circ} = 46^{\circ}$

  • Then, $\frac{\sin 20^{\circ}}{65} = \frac{\sin 114^{\circ}}{b}$, so $b = 65 \cdot \frac{\sin 114^{\circ}}{\sin 20^{\circ}}$

  • Similarly, $c = 65 \cdot \frac{\sin 46^{\circ}}{\sin 20^{\circ}}$

• Example 2 (SSA - Ambiguous Case): Given $a = 75$, $b = 100$, $A = 50^{\circ}$, solve for $B$:

  • $\frac{\sin 50^{\circ}}{75} = \frac{\sin B}{100}$

  • $\sin B = 100 \cdot \frac{\sin 50^{\circ}}{75}$

  • Check if $\sin B$ is between -1 and 1 to determine if a solution exists. If $\sin B > 1$, no solution; if $\sin B = 1$, one solution; if $0 < \sin B < 1$, one or two solutions (ambiguous case).

Summary Table: Triangle Solving Cases

Case	Given	Method	Notes
AAS	2 angles, 1 non-included side	Law of Sines	Unique solution
ASA	2 angles, included side	Law of Sines	Unique solution
SSA	2 sides, non-included angle	Law of Sines	Ambiguous case: 0, 1, or 2 solutions

Important Reminders

• Always check if the triangle is possible (e.g., the sum of any two sides must be greater than the third side).

• For ambiguous cases, consider all possible solutions.

• Use degree mode on your calculator when working with degrees.

Additional info:

• These notes are based on standard Precalculus curriculum topics, focusing on triangle geometry and the Law of Sines.