The Sine Rule

The sine rule is used to find out the length of a side or the angle of a triangle.  First you need to label the triangle like this...

the angles need to be labeled with capital letters and the lengths need to be labeled with lower case letters.  It does not matter which side is which as long as the angle opposite to a length is always the same letter as the length.

The sine rule is .  This is the one used to find out missing lengths.  

To use this rule you need 2 angles and 1 length.  For example you have A, B and a and I want to find length a.  EG.         A=45^o         B =30^o      b = 2cm

We know that angle A in the triangle above is 45^o, that angle B is 30^o and that the length b is 2 units.

To work out the remaining angle, we need to remember that the angles within a triangle always add up to 180^o.

Since we know A and B add up to 75^o, the angle C must be 105^o.

Now to find the length a, we can use the first part of the sine rule above. We can rearrange a/sinA = b/sinB to get a=bsinA/sinB.

Since we know A and B we can evaluate this expression to get 
a=

Finally we can use the second part of the sine rule to find the length c:

b/sinB=c/sinC, so c=bsinC/sinB

That gives c=2sin(105^o)/sin(30^o) 
which is 4sin(105^o).

We can write sin(105^o) as sin(150^o-45^o) then use the sin(A-B) rule to write this as 
sin(150^o)cos(45^o)-cos(150^o)sin(45^o)

Putting in the values for the sine and cosine of these special angles gives 
c=

It is quite simple, all you need to do is 

Indice rules

WHAT ARE INDICES?

Indices and numbers which are in index form such as...

3³ = 3×3×3
2¹ = 2
2² = 2×2
2³ = 2×2×2

There are certain rules which are used to work out how to simplify indices.   

These rules may look complex but if you follow the rules it is actually really simple.

For example what is 15² x 15³ ?  You don't even need to know what 15 squared or cubed is you just use the first rule.  and 2+3=5 so the answer is 15^5.  

Dividing is just the same apart from you subtract the powers.

Anything to the power of -1 inverses it.  You find what is called the reciprocal of it.  

For example the reciprocal of a quarter is 4 because it is the same as four over 1

1 4   to  4 1

^{Remember that anything to the power of 0 is always 1!}