# 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.

z^{2}=x^{2}+y^{2}-2xycosA.

If above triangle is right-angle triangle then A = 90 degree and above equation can be reduce to z^{2}=x^{2}+y^{2} 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,

X^{2} = Y^{2} + Z^{2} – 2YZcos(A)

Y^{2 }= X^{2} + Z^{2} – 2XZcos(B)

Z^{2} = X^{2} + Y^{2} – 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.

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

So in right-angle triangle ADC, b^{2} = h^{2} + x^{2} ------- (1)

cosA = x/b,

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

from right-angle triangle BDC,

a^{2} = (c-x)^{2} + h^{2}

^{ }so, a^{2} = x^{2} + h^{2} + c^{2} – 2cx ----(3)

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

a^{2} = b^{2} + c^{2} – 2(bc)cosA

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

b^{2} = a^{2} + c^{2} – 2(ac)cosB

c^{2} = a^{2} + b^{2} – 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.

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
**Let's see little bit difficult

**law of cosines example.**

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

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