Trigonometry is the study of triangles which connects sides to angles. It uses functions called the sine, cosine and tangent. You can find these on scientific calculators with buttons named sin, cos and tan.
Use the phrase SOHCAHTOA to help you remember how to use trigonometry:
SOH = Sin x = Opposite/Hypotenuse
CAH = Cos x = Adjacent/Hypotenuse
TOA = Tan x = Opposite/Adjacent
The most common use for these functions is to find sides and angles of a right angled triangle. The rules follow:

sin A° = ^{opposite}/_{hypotenuse} 
The sides and angles of a nonrightangled triangle can also be found using the following rules:

^{a}/_{sin A} =^{b}/_{sin B} = ^{c}/_{sin C }* You are able to turn the whole of this formulae upside down if it helps! a^{2} = b^{2} + c^{2}  2bc cos A 
The functions all give graphs which are important. You should know how to sketch them all and know how to use them.
The Sine Curve
x° 
0 
30 
60 
90 
120 
150 
180 
210 
240 
270 
300 
330 
360 
sin x° 
0.00 
0.50 
0.86 
1.00 
0.86 
0.50 
0.00 
0.50 
0.86 
1.00 
0.86 
0.50 
0.00 

This is the curve drawn when you put all the figures on the graph from the table above. As you can see, this curve is in a wave form. This wave can continue past 360° and go into the negatives. 
The Cosine Curve
x° 
0 
30 
60 
90 
120 
150 
180 
210 
240 
270 
300 
330 
360 
cos x° 
1.00 
0.86 
0.50 
0.00 
0.50 
0.87 
1.00 
0.87 
0.50 
0.00 
0.50 
0.87 
1.00 

If you look at this curve you can see it is actually the same as the sine curve except it is a different section (i.e. this peaks at 0° where the sine curve peaks at 90°). 
The Tan Curve
x° 
0 
30 
60 
90 
120 
150 
180 
210 
240 
270 
300 
330 
360 
tan x° 
0.00 
0.58 
1.73 

1.73 
0.58 
0.00 
0.58 
1.73 

1.73 
0.58 
0.00 

The tan curve is very different from the others. It is a noncontinuous which breaks as the value at the breaking point ( when x=90 or x=27) is infinity. Again this curve can be continued with the section from x=90 to x=270 repeated. 
From the curves we can see there is always more than one possible value for any number you are working out the inverse of ( sin^{1} 0.5 = 30° or 150° ). The problem is that your calculator only gives you one of the values ( the one below 90° ). You must remember the curves to find the position of the second angle.