A tangent, like a sine or a cosine, is a mathematical way of thinking about a kind of movement that happens in nature. A tangent is really just a rephrasing of the same information we already had for calculating the sine function.
One way to describe a tangent is that it is a line that just touches a circle at one single point: tangent comes from the Latin word tangere, to touch. Each point on a circle has only one line that can go through that point and not touch the circle anywhere else: that is the tangent line of that point. A plane can also be tangent to a sphere.
But another way to describe a tangent is related to the sine function: just as the sine function describes the relationship between the opposite side and the hypotenuse of a right triangle, and the cosine divides the adjacent side by the hypotenuse, the tangent divides the opposite by the adjacent sides. If you change the lengths of the sides of the triangle, the tangent changes along with the sine and the cosine. You can remember these three ratios with the mnemonic SOHCAHTOA: Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent.
The cosine wave and the sine wave for the same natural phenomenon go up and down by the same amount, but out of phase: the cosine reaches the bottom of the wave when the sine is at zero, and the sine reaches the bottom of the wave when the cosine is at zero. The tangent, on the other hand, is a straight line – not a wave – that moves around the edge of a circle defined by these right triangles, as you can see here.