Three Sides, One Angle
Part of the Triangles Without Right Angles section of Coddy's Trigonometry journey. Lesson 21 of 44.
With all three sides known, the law of cosines runs backwards and gives an angle. Write it for the angle you want: that angle is between the other two sides, and the side across from it goes on the left.
Sides 7, 8 and 13, and the angle φ across from 13: 13^2 = 7^2 + 8^2 - 2×7×8×cos(phi). That is 169 = 113 − 112 cos φ, so cos(phi) = -1/2 and φ = 120°.
On the board: pick Law of cosines, move the terms to get the cosine alone, then use arccos. Between 0° and 180° each cosine has exactly one angle, so there is no second case to test.
Challenge
A triangle has sides AB = 8, AC = 3 and BC = 7. Find the angle θ at A, exactly, then press Check.
Try it yourself
Use the figure to find theta
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Triangles Without Right Angles
2Law 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