The Law of Cosines
Part of the Triangles Without Right Angles section of Coddy's Trigonometry journey. Lesson 16 of 44.
Two sides and the angle between them (SAS) give no complete pair, so the law of sines is stuck. The law of cosines finds the third side, with C the angle between the sides a and b:
c^2 = a^2 + b^2 - 2×a×b×cos(C)
It is Pythagoras with a correction. At C = 90° the cosine is 0 and the last term vanishes. An angle under 90° squeezes the sides together, the cosine is positive and the term takes some length away.
Sides 8 and 15 with 60° between them: c^2 = 8^2 + 15^2 - 2×8×15×cos(60°), which is 64 + 225 − 120 = 169. c² = 169 gives c = 13 or c = −13, and a length is positive, so the side is 13. On the board the negative root is ruled out on its own.
Challenge
Two sides of a triangle are 16 cm and 10 cm, with an angle of 60° between them. Find the third side x exactly, then press Check.
Try it yourself
Use the figure to find x
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Triangles Without Right Angles
4Law of Cosines: SAS
The Law of CosinesAn Obtuse Angle BetweenThrough the HillA Hooked DriveRecap: Two Sides, One Angle2Law of Sines: AAS and ASA
Why the Law of Sines HoldsPair Side With AngleThe Third Angle FirstTwo Coast StationsRecap: Law of Sines8Bearings and Navigation
Reading BearingsTwo Ships From PortTurning Off CourseFinding a Course AngleRecap: Ships at Sea