06Zcosinerule

toc = The Cosine Rule =

Take __any__ triangle Then label
 * Label the angles ** A **, ** B **, ** C **
 * the side opposite angle ** A ** as side ** a **.
 * the side opposite angle ** B ** as side ** b **.
 * the side opposite angle ** C ** as side ** c **.

{Angle names are in capitals, side names are lower case}

This is the same labelling as for the Sine Rule

For this triangle, the Cosine Rule is:



Use the Cosine Rule when concerned with **3** sides and **1** angle - of which only 1 of these 4 things is unknown.

The Cosine Rule is symmetrical. It can be rewritten to involve any of the angles A, B or C.
 * Make the side opposite the angle, the subject of the equation (squared)
 * The other two sides then fill the roles on the right side of the equation

Finding a Side Length
{Before calculating the square root, keep the full version of 23.3839 on your calculator for greater accuracy}

NOTE If the unknown side is __not__ opposite the known angle, you will get a quadratic equation. Solve the equation using a CAS calculator or using the Quadratic Formula. You will usually get two answers as a result. (reject any negative answers)



{Notice that in this example, the two answers can be seen by swinging the side, c, up and down using the leftmost point of the triangle as a pivot.}

Finding an Angle




Also see the Sine Rule

Review Pythagoras' Theorem

For another site that explains this idea, go here: MathsIsFun

.