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The Law of Cosines - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

One method for solving for a missing length or angle of a triangle is by using the law of cosines. The law of cosines, unlike the law of sines, is similar to the Pythagorean theorem, but it works for all triangles, not just right triangles. In order to understand how to use the law of cosines effectively, one must understand the cosine function and inverse trigonometric functions.

Just like the law of sines, the law of cosines applies to any kind of triangle not just right triangles. And what it says is if I pick one of these angles let's say angle c, the side opposite of angle c is side c and what I can do is I can say that c squared is equal to the other 2 sides squared. So a squared plus b squared, so right now it kind of looks like the Pythagorean Theorem. And what the law of cosines really does is it adjusts the Pythagorean Theorem to fit any kind of a triangle. And that fitting is where we subtract 2 times a times b times the cosine of whatever angle you're talking about. In this case it's angle c, so I could write 3 different law of cosines for this one triangle. All depending on which angle I'm talking about.
What you're going to do on your homework or your quiz is you're going to have 3 of these 4 pieces of information, you're going to substitute in and you're going to solve. One other thing I should point out is if you just substitute in here for ab and the angle c you will not solve for c. What you would have to do is take the square root of both sides to find out what c is.

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