Suneil P. answered • 06/30/14

Knowledgeable and Passionate University of Pennsylvania Math Tutor

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:"

http://www.math.tamu.edu/~yasskin/calclab/DEMOS/trigdef/tan.html

Suneil P.

06/30/14