## Proof of Cosine Rule

Below is the proof by Pythagoras’s theorem of the cosine rule, a^{2}=b^{2}+c^{2}– 2bccosA.

This assumes you understand Pythagoras’s theorem (visit pythagoras’s theorm to view my lesson on it), how to use basic trigonometry(basic trigonometry lesson). If you want to learn how to use the cosine and sine rule, opposed to just learning the proof) visit by sine and cosine rule page.

The proof is done using the letters of the following triangle

and we are trying to prove the cosine rule:

**a ^{2}=b^{2}+c^{2}– 2bccosA**

**In triangle CBL**

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

a^{2} = c^{2} – 2cx + x^{2} + h^{2}

h^{2} = a^{2} -c ^{2}– x^{2} + 2cx *<<EQN1*

**in triangle CLA**

b^{2} = h^{2} + x^{2}

h^{2} = b^{2} – x^{2 }* <<EQN2*

*eqn1 – eqn2 ::* 0 = a^{2} – c^{2} – b^{2} +2cx

a^{2} = c ^{2}+ b^{2} – 2cx *<<EQN3*

** in CLA**

cosA = x/b

x = bcosA

**in eqn3**

**a ^{2} = c^{2} + b^{2} – 2bccosA**

So there is the proof for the cosine rule using pythagorases therom. If you found that usefull try looking at my other maths lessons

## Understand the Sine and Cosine Rules

This assumes you already have a knowledge of basic trigonometry(ir using sin, cos and tan in a right angled triangle, if you don’t click here to read my lesson on these) and aims to teach you how to use the sine and cosine rule.

In basic trigonometry you can only look at a right angled triangle which greatly limits its applications, however with these formula you can calculate sides and angles in any triangle provided you know enough information. They are proved by splitting one triangle in 1/2 so that the dividing line is perpendicular to one of the sides and therefore creating 2 right angled triangle in which the normal rules can be applied.

The following use symbols as defined in the above triangle. Note that side a is opposite angle A and b is opposite B etc

**Sine Rule**

a/sinA = b/sinB = c/sinC

This allows us to find both an angle and a side as we can invert all of the fractions and it remains true. This means if we know the side opposite the angle we want and any other side angle pair we can work out the angle we want, or we can work out a side if we know the angle opposite it and any other side angle pair.

EG)Lets say

a = 10cm

b = 5cm

B = 30^{o}

and we want to find angle A

we know a side angle pair, b and B, and we know the side opposite the angle we want so we can write the sine rule as

sinA / 10 = sinB/b >>note we don’t need to include the c parts as we dont know either c or C

sinA / 10 = sin30/5

sinA = 10sin30/5

sinA = 1

A = sin^{-1}1

A = 90^{o}

We can work out any angle or side in a similar way.

**Cosine rule**

This rule allows us to find an angle if we know all the sides or a side if we know the other 2 and a angle

c² = a² + b² – 2abcosC

To find an angle we can re-arrange it so

C = cos^{-1}((a^{2} + b^{2} – c^{2})/2ab)

Im sure you can put the numbers in yourself as ive show you how it can be written to find either an angle or side so ill leave you to it 🙂 enjoy

If you have any questions, improvements, or suggestions please leave a comment below or email me at woodford_4@hotmail.co.uk. Also visit my site at www.breakingwave.co.nr

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