The Law of Cosines: Applications and Problem Solving

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 sides in any triangle to the cosine of one of its angles. It is especially useful for solving triangles when the Law of Sines is not applicable, such as when given two sides and the included angle (SAS) or all three sides (SSS).

• Definition: The Law of Cosines states that for any triangle with sides a, b, and c, and corresponding opposite angles A, B, and C:

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

• This formula can be rearranged to solve for any side or angle as needed.

Application Example 1: Finding a Side

Suppose you are given two sides and the included angle of a triangle and need to find the third side. The Law of Cosines allows you to calculate the unknown side directly.

• Example: Given sides a and b, and included angle C, find side c:

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

• Plug in the known values for a, b, and C to solve for c.

Application: This is commonly used in navigation, surveying, and physics when direct measurement of a side is not possible.

Application Example 2: Finding an Angle

If all three sides of a triangle are known, the Law of Cosines can be rearranged to solve for an unknown angle.

• Formula for Angle:

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

• Take the inverse cosine to find the angle:

$C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right)$

• Example: Given sides a = 8, b = 7, c = 9, find angle C:

$C = \cos^{-1}\left(\frac{8^2 + 7^2 - 9^2}{2 \times 8 \times 7}\right)$

• Calculate the value inside the inverse cosine and then determine the angle.

Summary Table: Law of Cosines Applications

Given	Find	Formula
Two sides and included angle (SAS)	Third side	$c^2 = a^2 + b^2 - 2ab\cos(C)$
All three sides (SSS)	Any angle	$\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$

Key Points

• The Law of Cosines generalizes the Pythagorean Theorem for all triangles, not just right triangles.

• It is essential for solving triangles when the Law of Sines cannot be used.

• Always check that you are using the correct formula for the given information (SAS or SSS).

Additional info: The Law of Cosines is also useful in vector mathematics and physics for determining the magnitude of the resultant vector when two vectors are not perpendicular.