Question

Pi and trigonometry??

Answer 1

>what does pi have to do with trig

pi is the exact ratio of a circle's circumference to its diameter

​

>what are you actually doing when you take sin of a value, or cos or tan

On the unit circle, take any point on the circle's perimeter.  The radius extending to that point makes an angle θ with the x-axis.  For this angle θ, the sine of that angle is the y-coordinate of that point, while the cosine is the x-coordinate of that point.  The tangent is the sine divided by the cosine, and is the slope of the line the radius is part of.

Answer 2

Here's another way to look at it. If you take any point (x,y) on the unit circle, x is the cosine of the angle made by the line from the origin to (x,y) and the positive x-axis. And y is the sine.

So if you walk around the unit circle holding one of a string, the other end of which is tethered to the origin, the string will make an angle of theta radians with the positive x-axis. Then the coordinates (x,y) are actually (cos theta, sin theta).

From this you can recover SOHCATOA and the standard identities.