# What exactly is sin/cos/tan?

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I get this is kind of an overdone question, but I still don’t really understand what it is. I think of math visually, and for some reason, I just can’t really get an idea of what sin and cosine “look like,” just that they are ratios.

Thanks!

In: Mathematics

Although trig is supposedly the study of triangles, I think the unit circle picture is more intuitive for beginners. Basically, (cos(theta), sin(theta)) are the coordinates for the points of a circle.

OK, you’ll like this:

Get out a sheet of graph paper and draw the good old x and y axes with the origin in the middle. Now draw a circle with a radius of one. If you can, make the circle large, with ten little grids on the radius so that each grid is 0.1.

Okay, now start at the far right side of the circle, where x=1 and y=0. Now just go around the circle counterclockwise. You can stop at any point, and whatever the x value is, that is the cosine. Whatever the y value is, that is the sine. The angle is whatever the angle between the positive x axis and the line from the origin to the point is.

Let’s do some simple examples. Starting with the starting point, right there where x=1 and y=0. The angle is zero degrees, because we haven’t moved yet. And sure enough, the cosine of zero degrees is 1 and the sine is zero.

Now go 90 degrees around, to where x=0 and y=1. Hey look, the sine is 1 and the cosine is zero.

Now go halfway back down, to an angle of 45 degrees. At that point, x and y are the same, and by inspection you can see they are about 0.7. If you do the geometry with Pythagoras’ theorem, you will find that they are both the square root of two divided by two, which is 0.707… and sure enough that is both the sine and cosine of a 45 degree angle.

This works for any angle you try. As for the tangent, it is defined as the sine divided by the cosine. Sometimes that is useful.