The Law of Cosines: Solving Oblique Triangles

9.3 The Law of Cosines

Introduction

The Law of Cosines is a fundamental trigonometric formula used to solve oblique triangles, specifically when the triangle is not a right triangle. It is particularly useful for solving triangles when given two sides and the included angle (SAS case) or all three sides (SSS case). The Law of Cosines generalizes the Pythagorean Theorem to non-right triangles.

Law of Cosines: Definition and Formula

• Definition: For any triangle with sides a, b, and c opposite angles A, B, and C respectively, the Law of Cosines relates the lengths of the sides to the cosine of one of its angles.

• Formulas:

$a^2 = b^2 + c^2 - 2bc \cos A$ $b^2 = a^2 + c^2 - 2ac \cos B$ $c^2 = a^2 + b^2 - 2ab \cos C$

Alternate Form of the Law of Cosines

• Solving for the cosine of an angle:

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

When to Use the Law of Cosines

• Use the Law of Cosines when:

  • Given two sides and the included angle (SAS case).

  • Given all three sides (SSS case).

  • The Law of Sines cannot be directly applied due to lack of sufficient angle-side pairs.

Solving Oblique Triangles Using the Law of Cosines

Solving the SAS Case

When two sides and the included angle are known, the Law of Cosines allows you to find the third side and then use the Law of Sines or Cosines to find the remaining angles.

• Step 1: Use the Law of Cosines to find the third side.

• Step 2: Use the Law of Sines or Cosines to find one of the remaining angles.

• Step 3: Use the fact that the sum of the angles in a triangle is $180^\circ$ to find the last angle.

Example: Given $a = 7$, $b = 5$, and $C = 60^\circ$, find side $c$. $c^2 = a^2 + b^2 - 2ab \cos C$ $c^2 = 7^2 + 5^2 - 2 \times 7 \times 5 \times \cos 60^\circ$ $c^2 = 49 + 25 - 70 \times 0.5 = 74 - 35 = 39$ $c = \sqrt{39} \approx 6.24$

Solving the SSS Case

When all three sides are known, the Law of Cosines can be used to find each angle.

• Step 1: Use the alternate form of the Law of Cosines to find the measure of the largest angle.

• Step 2: Use the Law of Sines or Cosines to find the other angles.

• Step 3: Use the triangle angle sum property ($180^\circ$) to find the remaining angle.

Example: Given $a = 8$, $b = 6$, $c = 5$, find angle $A$. $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$ $\cos A = \frac{36 + 25 - 64}{2 \times 6 \times 5} = \frac{-3}{60} = -0.05$ $A = \cos^{-1}(-0.05) \approx 92.87^\circ$

Solving Applied Problems Involving Oblique Triangles

The Law of Cosines is frequently used in real-world applications such as navigation, surveying, and physics, where triangle measurements are required but the triangle is not a right triangle.

• Use the Law of Cosines to determine unknown distances or angles in applied scenarios.

• Always check if the triangle is oblique (no right angle) before applying the Law of Cosines.

Example: In navigation, if two ships leave a port at different angles and speeds, the Law of Cosines can be used to determine the distance between them after a certain time.

Summary Table: Law of Cosines Applications

Case	Given	Find	Method
SAS	Two sides and included angle	Third side, remaining angles	Law of Cosines, then Law of Sines
SSS	All three sides	All angles	Law of Cosines
Applied Problems	Various measurements	Unknown sides or angles	Law of Cosines

Additional info: The Law of Cosines is essential for solving triangles that do not contain a right angle and is a key tool in trigonometry for both theoretical and applied problems.