George H.

# What is the tangent function?

I'm not asking what it does, and I'm not looking for "tan(angle)=opposite leg/adjacent leg". I'm asking what the actual function is. For instance, using degrees (not radians) tan(22.5)=20.5 - 1. What equation is 22.5 plugged into to get approximately 0.4142? tan(45)=1. What equation is 45 plugged into to get 1? I tried searching this on google but of course everything was "The tangent function is used for...." or "the tangent function equals the opposite over adjacent...." so I figured my best bet was here. Thanks!

PS: I'm pretty sure it uses integrals so it would be nice if you explained how those specific ones work out. Thanks again!

By: Knowledgeable and Passionate University of Pennsylvania Math Tutor Suneil P.

When our angle is, for instance, 0, our hypotenuse is flat along the x-axis...and so our slope is 0...so tan(0)=0.

When our angle is, for instance, PI/4 or 45 degrees, our rise=run and so slope is 1, making tan(45 deg)=1

when our angle is 90 deg or PI/2, our "hypotenuse" is vertical with an undefined slope, and hence tan(90 deg)=undefined
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06/30/14

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