Law of Cosines

Learning Objectives State the three standard forms of the Law of Cosines. Identify when to apply the Law of Cosines (SAS and SSS cases). Use the Law of Cosines to calculate an unknown side length of an oblique triangle. Rearrange the Law of Cosines to calculate an unknown angle measure. Solve multi-step, real-world problems by applying the Law of Cosines. Distinguish between problems requiring the Law of Sines versus the Law of Cosines. How can a surveyor measure the distance across a canyon without ever crossing it? The Law of Cosines gives us the tools to solve triangles that aren't right-angled! 📐 This tutorial introduces the Law of Cosines, a powerful extension of the Pythagorean theorem for any triangle. You will learn how to find missing sides and angles in obli...

TermDefinitionExample Oblique TriangleA triangle that does not contain a right angle (90°). It can be either acute (all angles less than 90°) or obtuse (one angle greater than 90°).A triangle with angles 50°, 60°, and 70° is an oblique triangle. Included AngleThe angle formed between two specified sides of a triangle.In triangle ABC, angle C is the included angle between side 'a' and side 'b'. SAS (Side-Angle-Side) CaseA scenario where you know the lengths of two sides and the measure of the included angle of a triangle.Given side b = 10 cm, side c = 12 cm, and angle A = 45°. This is an SAS case, perfect for the Law of Cosines. SSS (Side-Side-Side) CaseA scenario where you know the lengths of all three sides of a triangle.Given side a = 5, side b = 7, and side c = 8. T...

Law of Cosines (Solving for a Side) 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) Use this form when you know two sides and the included angle (SAS) and need to find the side opposite the known angle. Notice the pattern: the side you are solving for matches the angle used in the cosine function. Law of Cosines (Solving for 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} Use this rearranged form when you know all three sides (SSS) and need to find an angle. To find the angle itself, you must use the inverse cosine function (cos⁻¹) on the result.