##### Like what you saw?

##### Create FREE Account and:

- Watch all FREE content in 21 subjects(388 videos for 23 hours)
- FREE advice on how to get better grades at school from an expert
- FREE study tips and eBooks on various topics

# Graph of the Tangent Function - Concept

###### 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.

Please enter your name.

Are you sure you want to delete this comment?

###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

Thiswas EXCELLENT! I am a math teacher and have been looking for an easy/logical way to explain the lateral area of a cone to my students and this was incredibly helpful, thank you very much!”

I just learned more In 3 minutes of polygons here than I do in 3 weeks in my math class”

Hahaha, his examples are the same problems of my math HW!”

##### Concept (1)

#### Related Topics

- Graphs of the Sine and Cosine Functions 38,145 views
- Transforming the Graphs of Sine and Cosine 31,568 views
- More Transformations of Sine and Cosine 16,650 views
- Find an Equation for the Sine or Cosine Wave 50,818 views
- The Tangent Function 18,958 views
- Evaluating the Tangent Function 12,139 views
- Transforming the Tangent Graph 20,362 views
- Intercepts and Asymptotes of Tangent Functions 28,835 views
- Trigonometric Identities 22,098 views
- The Reciprocal Trigonometric Functions 33,479 views
- Graphing the Reciprocal Trigonometric Functions 15,266 views
- Using Trigonometric Identities 31,827 views
- Transforming the Cotangent Graph 16,865 views
- Transforming Secant and Cosecant 19,553 views
- Asymptotes of Secant, Cosecant, and Cotangent 31,364 views
- Radian Measure of Angles 33,818 views
- The Definitions of Sine and Cosine 21,550 views
- Evaluating Sine and Cosine at Special Acute Angles 19,261 views
- Evaluating Sine and Cosine at Other Special Angles 21,074 views

## Comments (0)

Please Sign in or Sign up to add your comment.

## ·

Delete