Graph of the Tangent Function - Concept
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.
Please enter your name.
Are you sure you want to delete this comment?
- Graphs of the Sine and Cosine Functions 38,480 views
- Transforming the Graphs of Sine and Cosine 31,838 views
- More Transformations of Sine and Cosine 16,862 views
- Find an Equation for the Sine or Cosine Wave 51,794 views
- The Tangent Function 19,227 views
- Evaluating the Tangent Function 12,310 views
- Transforming the Tangent Graph 20,588 views
- Intercepts and Asymptotes of Tangent Functions 29,187 views
- Trigonometric Identities 22,343 views
- The Reciprocal Trigonometric Functions 33,966 views
- Graphing the Reciprocal Trigonometric Functions 15,463 views
- Using Trigonometric Identities 32,171 views
- Transforming the Cotangent Graph 17,060 views
- Transforming Secant and Cosecant 19,798 views
- Asymptotes of Secant, Cosecant, and Cotangent 31,884 views
- Radian Measure of Angles 34,224 views
- The Definitions of Sine and Cosine 21,945 views
- Evaluating Sine and Cosine at Special Acute Angles 19,592 views
- Evaluating Sine and Cosine at Other Special Angles 21,391 views