Exploring the Law of Cosines: Applications and Derivations

The Law of Cosines is a powerful tool in trigonometry, extending the Pythagorean Theorem to oblique triangles. This article will delve into its applications, derivation, and compare it with the Sine Law. Let's start with the basic expression of the Law of Cosines.

Understanding the Law of Cosines

The Law of Cosines states that for any triangle with sides of length a, b, and c, where A is the angle between sides b and c, the following relationship holds:

a^2 b^2 c^2 - 2bc cdot cos(A)

This equation generalizes the Pythagorean Theorem, which is a special case of the Law of Cosines where the angle A is 90 degrees and cos(90°) 0.

Geometric Interpretation and Proof

To understand the geometric interpretation, consider a right triangle ABC on the xy-plane, with C at the origin (0,0), A on the positive x-axis such that AC b, and B somewhere in quadrant I or II such that BC a and angle ACB theta. Let AB c.

Applying the distance formula to find the length c, we get:

[c sqrt{(a cos(theta) - b)^2 (a sin(theta))^2}]

Squaring both sides, we obtain:

[c^2 (a cos(theta) - b)^2 (a sin(theta))^2]

Expanding the equation:

[c^2 a^2 cos^2(theta) - 2ab cos(theta) b^2 a^2 sin^2(theta)]

Using the Pythagorean identity sin^2(theta) cos^2(theta) 1, we simplify:

[c^2 a^2 (sin^2(theta) cos^2(theta)) - 2ab cos(theta) b^2]

[c^2 a^2 - 2ab cos(theta) b^2]

[c^2 a^2 b^2 - 2ab cos(theta)]

Practical Applications of the Law of Cosines

The Law of Cosines finds applications in various fields, including engineering, surveying, and physics. Specifically, it allows you to find the length of an unknown side of a triangle if you know the lengths of the other two sides and the included angle.

Comparison with the Sine Law

While the Law of Cosines deals with sides and their included angle, the Sine Law (also known as the Law of Sines) is used to find unknown angles or sides when two angles and one side are given, or two sides and an angle opposite a given side are known. The Sine Law states:

A / sin(A) B / sin(B) C / sin(C)

Both laws are essential but serve different purposes in solving triangles.

Deriving the Formulae

The Law of Cosines can be derived using the geometric method as shown. Another approach involves vector algebra, trigonometric identities, and the distance formula. Here's a step-by-step derivation:

Step 1: Starting with the Distance Formula

We use the distance formula for the length of side c in a coordinate system, similar to how we defined the triangle in the previous section: