Sine and Cosine use circles to describe the lengths of the sides of triangles!

Imagine you were at the center of a circle, and you wanted to walk to a point on the edge of the circle. You could turn to it and walk in a straight line, making no turns or other movements. That's not the only way you could have walked to that point the circles edge!

You could have chosen one direction and walked forward until you got as close as possible to the place on the edge of the circle you were trying to get to. Then, you could have turned 90 degrees, towards your destination, and walked forward until you reached it.

Those two distances you walked are the cosine and the sine, scaled by the radius of the circle.

We usually fix the first direction you would walk in and call it the 'x' direction, and likewise the second direction would be the 'y' direction. We'd call the first distance the cosine, in that case, and the second direction the sine.

Because these two lengths are perpendicular, meaning they are at 90 degree angles to each other, they form a right triangle. The hypotenuse of this triangle is the distance you walked when you just picked a point and walked straight to it, and the two sides are the sine and cosine, scaled by the length of the hypotenuse.

The exact relationship between the two is that they are essentially "90 degrees apart" from one another. They both follow the same pattern: you could have easily switched the directions you walked in earlier and gotten to the same place! Because of this, we say they are the same function, but "90 degrees out of phase."

I hope that helped! Is there anything else I could clarify?
spam fails_task lang_mismatch pii not_appropriate hate_speech sexual_content quality toxicity humor helpfulness creativity violence