Can someone tell my what cos, sin and tan actually measure?
In: Mathematics
I think the unit circle helps to understand a lot of the ideas behind those, and why they come up so often. [https://en.wikipedia.org/wiki/Unit_circle](https://en.wikipedia.org/wiki/Unit_circle)
Everyone is starting explaining stuff with triangles. To me, it’s a lot more obvious looking at this circle. it makes most of the formulas and theorem around them get pretty obvious, instead of something arbitrary you have to learn from memory.
You draw a circle with radius 1. For any given point on that circle, you look at the angle it makes with the horizontal. And then, cosines and sines of that angle are the coordinates of that point on the circle. They are just telling you where you’ll cross the circle if you draw a line starting from the center, and going at that angle.
Granted, Tan is a bit more tricky.
Example of stuff that gets obvious:
_ Sin and cos are always between -1 and 1. Well, the circle just has radius 1, so clearly no point goes further than that.
_ For any x, cos²(x) + sin²(x) = 1. Looks like a weird formula ? That’s just looking at the radius of the circle, putting the coordinates in the formula for distance between the point on the circle and the center. We know that distance is 1, that was the definition.
_ cos is the ratio of adjacent side of a triangle to hypotenuse. You can just draw that triangle in the circle. cosines is the length of that side. Then the division by hypotenuse is fixing the scale, since in our unit circle, hypotenuse is 1. Same for sin.
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