Law of Cosines

Learning Objectives State the three standard forms of the Law of Cosines. Use the Law of Cosines to calculate an unknown side length of an oblique triangle given two sides and the included angle (SAS). Use the Law of Cosines to calculate any angle of a triangle given all three side lengths (SSS). Algebraically rearrange the Law of Cosines formula to isolate the cosine of an angle. Identify when it is appropriate to apply the Law of Cosines versus the Law of Sines. Solve multi-step problems and real-world applications involving the Law of Cosines. Recognize the Law of Cosines as a generalization of the Pythagorean Theorem. How can a surveyor determine the length of a lake without ever crossing it, just by measuring two distances from a single point and the angle between the...

TermDefinitionExample Oblique TriangleAny 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. SAS (Side-Angle-Side) CaseA scenario in which the lengths of two sides and the measure of the included angle are known. The Law of Cosines is used to find the third side.Given side b = 10 cm, side c = 12 cm, and the angle between them, A = 45°. SSS (Side-Side-Side) CaseA scenario in which the lengths of all three sides of a triangle are known. The Law of Cosines is used to find any of the angles.Given side a = 7 m, side b = 9 m, and side c = 10 m. Included AngleThe angle formed at the vertex where two specific sides of a triangle meet.In triangl...

Law of Cosines (Solving for a Side) a² = b² + c² - 2bc cos(A) b² = a² + c² - 2ac cos(B) c² = a² + b² - 2ab cos(C) Use this form when you know two sides and the included angle (SAS) and need to find the length of the side opposite the known angle. Law of Cosines (Solving for an Angle) cos(A) = (b² + c² - a²) / 2bc cos(B) = (a² + c² - b²) / 2ac cos(C) = (a² + b² - c²) / 2ab Use this rearranged form when you know all three sides (SSS) and need to find the measure of an angle. After calculating the value of the cosine, use the inverse cosine function (cos⁻¹) to find the angle.