An Obtuse Angle Between
Part of the Triangles Without Right Angles section of Coddy's Trigonometry journey. Lesson 17 of 44.
When the angle between the sides is obtuse, its cosine is negative, so the term -2×a×b×cos(C) adds length instead of taking it away. The third side comes out longer than Pythagoras would give.
Sides 5 and 8 with 120° between them: cos(120°) = -1/2, so the last term is -2×5×8×(-1/2), which is +40. Then c² = 25 + 64 + 40 = 129, and c = √129, about 11.36. Pythagoras alone would have given √89, about 9.43: too short for a wide-open angle.
The usual mistake is to take cos 120° as +1/2, which subtracts 40 and gives √49 = 7, a side shorter than the 8 next to it. An obtuse angle faces the longest side, so that answer cannot be right.
Challenge
Two sides of a triangle are 7 m and 8 m, and the angle between them is 120°. 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