Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Trigonometry - non-right angled triangles.
Cosine Rule - introduction.


 

As noted previously, the Cosine Rule is an equation linking three angles and one side in a triangle. Any question involving the applcation of the Cosine Rule will give you three of these four measurements and ask you to determine the fourth - whether that be the angle or one of the sides.

The proof of the relationship amongst the sides and the angle - so the "formula" - is as follows:

Draw a triangle.

Mark the three vertices as A, B and C.

Label the three sides using the lower case letter from the angle opposite.
(This is just a convention so as to obtain the result in a common format).

Drop a vertical from the apex perpendicular to the side opposite and mark its height as h.

Mark the distance from vertex A to the vertical with the distance x and so the balance of that side has the length of (b - x).
Write down the Pythagorean equations for the two triangles with the right angles using h as the subject:
Equate the two equations and simplify to make x the subject.
Write down the usual equation for cos A, substitute and simplify