The Law of Cosines in Trigonometry

The Law of Cosines

Introduction to the Law of Cosines

The Law of Cosines is a fundamental theorem in trigonometry used to relate the lengths of the sides of a triangle to the cosine of one of its angles. It is especially useful for solving triangles when two sides and the included angle are known, or when all three sides are known.

• Definition: The Law of Cosines generalizes the Pythagorean Theorem to all types of triangles, not just right triangles.

• Formula: For any triangle with sides a, b, and c, and corresponding opposite angles A, B, and C:

Law of Cosines Equations:

• $c^2 = a^2 + b^2 - 2ab\cos(C)$

• $a^2 = b^2 + c^2 - 2bc\cos(A)$

• $b^2 = a^2 + c^2 - 2ac\cos(B)$

Solving Triangles Using the Law of Cosines

To solve for an unknown side or angle in a triangle:

1. Identify the known sides and angles.

2. Substitute the known values into the appropriate Law of Cosines formula.

3. Solve for the unknown side or angle.

Example Problem

Given: Triangle with sides a = 7, b = 9, and included angle C = 60^b0. Find side c.

• Apply the Law of Cosines:

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

$c^2 = 7^2 + 9^2 - 2 \times 7 \times 9 \times \cos(60^b0)$

$c^2 = 49 + 81 - 126 \times 0.5$

$c^2 = 130 - 63$

$c^2 = 67$

$c = \sqrt{67} \approx 8.19$

Applications

• Solving for unknown sides or angles in oblique triangles (non-right triangles).

• Navigation, engineering, and physics problems involving non-right triangles.

Comparison: Law of Sines vs. Law of Cosines

Law of Sines	Law of Cosines
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$	$c^2 = a^2 + b^2 - 2ab\cos(C)$
Used when two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) are known.	Used when two sides and the included angle (SAS) or all three sides (SSS) are known.

Additional info: The handwritten notes on the image show a worked example using the Law of Cosines to solve for a side, consistent with the above explanation.