Why do we use trigonometry and how to we get the sine, cosine, and tangent graphs?

The three trigonometric functions sine, cosine, and tangent are derived from looking at the ratios of side lengths in a right-angled triangle. (Triangle drawn and side lengths annotated H=hypotenuse, A=adjacent side, O=opposite side, theta drawn for angle).For a given angle theta, sin(theta) tells us the ratio between the opposite and hypotenuse lengths, cos(theta) the ratio between the adjacent and hypotenuse lengths, and tan(theta) the ratio between the opposite and adjacent lengths. You can then show using algebra (formulae written out) that tan(theta)=sin(theta)/cos(theta).
To understand better why we get negative values for sin, cos, and tan, I will introduce the unit circle. This is a circle on the x-y plane, centred at the origin, with radius 1. By drawing an appropriate right-angled triangle in the circle, you can see how sin and cos, and therefore tan in turn, change with theta (much more easily demonstrated through diagram). If we plot these values against theta for sin and cos, we get the well-known sin and cos curves, and by knowledge of fractions we can plot out the tan curve.
At GCSE level, trigonometry is used mostly to work out angle and side lengths in right angled triangles, although this will be extended to irregular triangles through means such as the sine and cosine rule. Trigonometry is seen much throughout all of maths, through A-Level and University, and it is essential to have a grasp of what is going on from an early stage.