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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!

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Suneil P. | Knowledgeable and Passionate University of Pennsylvania Math TutorKnowledgeable and Passionate University ...
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No need integrals :)
The tangent function comes from what you said: opp/adj
Imagine a unit circle on a cartesian plane.  Now, let us suppose we are working with angles in Quadrant I; suppose we are working with some angle Θ; then we can form a right triangle corresponding to this angle (its three segs are the segment connecting the origin to a point on the circle that is Θ counterclockwise with respect to the x-axis, and then the segment drawn from that point on the circle perpendicularly down to the x-axis, and finally the segment from the origin to this point on the x-axis).  The first of the three segs is along what we call the terminal ray and forms the hypotenuse of the said triangle.  The segment along the x-axis is part of what we refer to as the initial ray.
Then tan(Θ) is simply opp/adj or the length of the side opposite the angle divided by the length of the side adj to it.  Graphically, this is just the y-coordinate divided by the x-coordinate of the terminal point (the intersection of the hypotenuse and the circle).  This is just the slope of the hypotenuse.
To graph the function, we plot the tan values for different input angle values of theta.
For a nice illustration of how the tan function is formed, this may help:"


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