What exactly is sin/cos/tan?

726 views
0

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

They are trigenomatric functions. Given one of the angles of a right angled triangle they give you the ratio of the lengths of two of the sides.

I don’t have it right now and this’ll probably get deleted because it’s not really an answer but if you google “trig gif” or “sin/cosine gif” there are some really good visualizations that show the different ratios and relationships in trigonometry. There’s also, /r/trigonometry and /r/geometry, they have some stuff like that.

If you have a right triangle, and you choose an angle that is not the 90 degree angle, sine is the ratio of the line opposite that angle to the hypotenuse. When I say ratio, think you literally measure how long the line opposite is, then divide it by how long the hypotenuse is. It’s a number your get by taking the ratio (think fraction) of the lengths of certain sides of the triangle. Soh Cah Toa. Sin(angle) = opposite length/hypotenuse length. It’s not a thing by itself if that’s what is messing you up. It’s hard to do an ELI5 on this without getting some things a little wrong via the simplicity. If you really want to understand, plug in a bunch of values from small angles to big angles for sine and see what happens. I think that’ll help you visualize it.

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.

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.

https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:trig/x2ec2f6f830c9fb89:unit-circle/a/trig-unit-circle-review