The Law of Cosines

[Audio] Hi everyone! In this short video, I'll explain the Law of Cosines, a key concept from Module 7. This formula helps us solve oblique triangles when we know two sides and the included angle (SAS) or all three sides (SSS)..

[Audio] Overview of the Law of Cosines. Introduction to the Law of Cosines.

[Audio] The Law of Cosines is a fundamental concept in trigonometry that extends the Pythagorean Theorem to apply to all types of triangles, not just right triangles. It provides a relationship between the lengths of the sides of a triangle and the cosine of one of its angles. This law is particularly useful for solving oblique triangles—those that do not contain a right angle—when certain combinations of sides and angles are known. The formula is expressed as: c squared equals a squared plus b squared minus 2 times a b times cos (C), where a, b, and c are the sides of the triangle and uppercase A, B, and C are the angles opposite sides. This formula allows for the calculation of unknown sides or angles, making it a powerful tool in geometry and trigonometry..

[Audio] Step-by-step example. Applying the Law of Cosines.

[Audio] To illustrate the Law of Cosines, consider a triangle with sides a equals 7, b equals 9, and c equals 10. The goal is to find angle C, which is opposite side c. First, apply the Law of Cosines formula: and Substituting the given values, we get cos(C) = (49 plus 81 minus 100) over (2 times 7 times 9) equals 30 over 126 which is approximately 0.2381. Next, use the inverse cosine function to find the angle: C = cos to the negative power 1 (0.2381) is approximately 76.2°. This example demonstrates how the Law of Cosines can be used to determine an angle when all three sides of a triangle are known. It is a straightforward application of the formula and highlights the utility of trigonometric principles in solving geometric problems..

[Audio] Diagram of triangle and formula. Visual Representation.

[Audio] A visual representation of the Law of Cosines involves a triangle labeled with vertices A, B, and C. Side c is opposite angle C and lies between points A and B. Sides a and b extend from point C to points B and A, respectively. The angle C is located at vertex C, between sides a and b. The diagram includes the formula c squared equals a squared plus b squared minus 2 times a b times cos (C). To reinforce the relationship between the triangle's sides and angle. This visual aid helps in understanding how the formula is derived and applied, making the concept more accessible and easier to grasp for learners..