Details on law of cosine

Law of cosine is a generalized Pythagoras theorem. Because it relates the two sides of any types of triangle and angle between this two sides to the third side of that triangle. Suppose x, y and z are three side of a triangle which is not right-angle triangle, and A is the angle between x & y then according to law of cosines third side z is given as below.


If above triangle is right-angle triangle then A = 90 degree and above equation can be reduce to z2=x2+y2 which is Pythagoras theorem.

Law of cosines formula Description

Law of cosines formula relates all three sides of any types of triangle and a respected angle. Below are some applications of cosine law formula.

1. If you know all three sides length, then you can easily find angle between any pair of sides.

2. If you know length of two sides and a angle then you can easily find length of third side and remaining angles for any types of triangle.

3. If X,Y, and Z are three sides of a any triangle and A, B and C are respected angles between them, then law of cosines formula is given as,

X2 = Y2 + Z2 – 2YZcos(A)

Y2 = X2 + Z2 – 2XZcos(B)

Z2 = X2 + Y2 – 2XYcos(C)

When we see law of cosine example, the significance of above equations more cleared.

Deriving law of cosines proof

For deriving law of cosines proof assume that a triangle have sides a, b, c and angles A, B and C.
deriving law of cosines proof

From angle C put perpendicular D on side c as shown in figure.

So in right-angle triangle ADC, b2 = h2 + x2 ------- (1)

cosA = x/b,

so, x = bcosA --------(2)

from right-angle triangle BDC,

a2 = (c-x)2 + h2

so, a2 = x2 + h2 + c2 – 2cx ----(3)

putting value of (1) and (2) in equation (3),

a2 = b2 + c2 – 2(bc)cosA

In same manner we can prove law of cosines for another two angles.

b2 = a2 + c2 – 2(ac)cosB

c2 = a2 + b2 – 2(ab)cosB

Above law is for two dimensions, spherical law of cosines also works for three dimensions. It relates the arcs and angles of spherical triangles. Assume that a,b,c are sides and A, B, C are angles of spherical triangles as shown in below figure.
spherical law of cosines

For this cos(c) = cos(a)cos(b) + sin(a)sin(b)sin(C)

similarly, cos(A) = -cos(B)cos(C) + sin(B)sin(C)cos(a)

Now we solved some law of cosines examples to get idea about application of law of cosines.

Law of cosines examples with solution
examples of law of cosines

Let's see little bit difficult law of cosines example.

law of cosines examples with solution

Now we see example for law of cosines word problem.

Solution for law of cosines word problem

Law of cosines word problems can be solved as shown below

law of cosines examples

Ian Roberts
San Francisco, USA

"If you're at school or you just deal with mathematics, you need to use This thing is really helpful."
Lisa Jordan
Math Teacher
New-York, USA

"I will recommend Studygeek to students, who have some chalenges in mathematics. This Site has bunch of great lessons and examples. "
John Maloney
Student, Designer
Philadelphia, USA

" I'm a geek, and I love this website. It really helped me during my math classes. Check it out) "
Steve Karpesky
Vancuver, Canada

"I use a lot on a daily basis, helping my son with his geometry classes. Also, it has very cool math solver, which makes study process pretty fun"