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Graph of the Tangent Function - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

For a tangent function graph, create a table of values and plot them on the coordinate plane. Since tan(theta)=y/x, whenever x=0 the tangent function is undefined (dividing by zero is undefined). These points, at theta=pi/2, 3pi/2 and their integer multiples, are represented on a graph by vertical asymptotes, or values the function cannot equal. Because of unit circle symmetry over the y-axis, the period is pi/2.

I want to graph the tangent function. I have a table of values written here and the definition of the tangent function on the unit circle here. Now here's the unit circle. I want to remind you that another way to see the tangent function as the slope of the terminal side op. Why is that? Well it's because you draw this little triangle here, the vertical leg of the triangle is y and the horizontal leg is x where x and y are these coordinates. And the slope of this line would be y over x rise over run. So y over x is the slope of op and that kind of helps us see how tangent behaves. But tangent gives me the slope of this line.
Alright. Let's start by plotting some points, I'll come back to the slope issue in a second. The first point is 0 0, that goes right there. And I'm just going to use these 2 points. Pi over 4, 1. Pi over 4 is halfway between 0 and pi over 2, so right here. And I'm going to make this 1. So here is pi over 4, 1. And then pi over 3, root 3. Root 3 is approximately 1.7, so I'm going to plot that as 1.7, and pi over 3 is two thirds the way from 0 to pi over 2. So this is pi over 3 right there. Okay. If that's 1.5 and that's 2, 1.7 is about here. So there's my point and I draw my curve. It increases very rapidly like that and it actually has a vertical asymptote. It just increases steep more steeply and steeply as x approaches or as theta rather approaches pi over 2. And the reason for that is again it comes back to slope. As this angle gets closer and closer to pi over 2, the slope of this line gets steeper and steeper. It's approaching infinity and that's why the tangent zooms off to infinity.
So know this graph because in a future episode, we're going to extend this in both directions because tangent's actually defined for all real numbers.

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