Prove Law of Sines and Law of Cosines

Common Core: HSG-SRT.D.10

Law of Sines

Law of Sines: (sin A)/a = (sin B)/b = (sin C)/c

Proof: Law of Sines.

Trig: Law of Sines - The Derivation.

The Law of Sines
The Law of Sines is a relationship among the angles and sides of a triangle. The ratio of the sine of any of the interior angles to the length of the side opposite that angle is the same for all three interior angles.

Law of Cosines

Law of Cosines: c² = a² + b² - 2abcosC

The law of Cosines is a generalization of the Pythagorean Theorem. If angle C were a right angle, the cosine of angle C would be zero and the Pythagorean Theorem would result.

Law of cosines
A proof of the law of cosines using Pythagorean Theorem and algebra.



The Law of Cosines - Proof
This is a proof of the Law of Cosines that uses the xy-coordinate plane and the distance formula. It does not introduce any letters other than a, b, c, and ?. The idea is that we move a triangle such that one of the sides rests on the x-axis; the formula comes from algebraic manipulation after finding the length of the side opposite the angle. It also works for any angle, so we don't have to do tedious proofs for acute angles, obtuse angles, and angles greater than 180 degrees.
(Errata: b²sin? should be b²sin²?)

Check out our most popular games!

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.