sine, cosine and tangent
Imagine standing at the foot of a ramp. The steeper the ramp, the more you climb for each step forward. Sine, cosine, and tangent are the three numbers that turn an angle into exactly that kind of relationship between heights and widths. They are the most-used tools in all of trigonometry, and they begin life as simple ratios of the sides of a right triangle.
Take a right triangle and fix on one of its acute angles, call it theta. Relative to theta the three sides have names: the hypotenuse (the longest side, opposite the right angle), the opposite side (across from theta), and the adjacent side (next to theta). Then sin theta = opposite / hypotenuse, cos theta = adjacent / hypotenuse, and tan theta = opposite / adjacent. Notice tan theta = sin theta / cos theta, since the hypotenuse cancels. A worked case: in a 3-4-5 right triangle, if the side opposite theta is 3 and the adjacent is 4 (so the hypotenuse is 5), then sin theta = 3/5 = 0.6, cos theta = 4/5 = 0.8, and tan theta = 3/4 = 0.75.
The reason these ratios are useful is that they depend only on the angle, not on the size of the triangle: any two right triangles with the same acute angle theta are similar, so the ratios of corresponding sides match. That single fact lets a calculator store one value of sin theta for every angle and lets us measure unreachable heights and distances. Later the definitions are widened to all angles using the unit circle, but the right-triangle picture is where the meaning lives.
A 12-metre ladder leans against a wall, making a 70 degree angle with the ground. How high up the wall does it reach? The wall height is the side opposite the 70 degree angle, with the ladder as hypotenuse, so height = 12 x sin 70 degrees, which is about 12 x 0.940 = 11.3 metres.
Choosing the ratio that links the side you know (hypotenuse) to the side you want (opposite) picks out sine.
Sine and cosine are always between -1 and 1, but tangent is unbounded and is undefined at 90 degrees, where cos theta = 0 and you would divide by zero — a frequent source of calculator errors.