Pi and trigonometry??

>what does pi have to do with trig

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

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>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.
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.
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