Symmetric About the Origin

[harpazo]

Cohen stated:

"First, the identity tan(-x) = - tan x tells us that the graph of y = tan x is symmetric about the origin. So, after reflecting the graph about the origin, we can draw the graph of y = tan x on the interval (-pi/2, pi/2) as shown in the figure."

Am I right for saying that y = tan x has an asymptote at -pi/2 and pi/2?

I do no understand what Cohen means by introducing tan(-x) = - tan x.



He then goes on to say the following which makes no sense to me. In this picture, Cohen introduces the unit circle. I do not see the connection between the unit circle and the tangent function as used here.



I think it is easier for me just to watch video clips on You Tube showing how to graph the trig functions. At the same time, I would like to understand my textbooks. Otherwise, it makes no sense to have math textbooks in my small library of books.

 

[Dr.Peterson]

Cohen stated:

"First, the identity tan(-x) = - tan x tells us that the graph of y = tan x is symmetric about the origin. So, after reflecting the graph about the origin, we can draw the graph of y = tan x on the interval (-pi/2, pi/2) as shown in the figure."

Am I right for saying that y = tan x has an asymptote at -pi/2 and pi/2?

Yes, it has asymptotes x = -pi/2 and x = +pi/2, as seen in the graph. There are more at x = 3pi/2, 5pi/2, and so on; in general, any odd multiple of pi/2 (that is, ..., -5, -3, -1, 1, 3, 5, ... times pi/2). The distance from any of these to the next is pi, the period of the tangent function.

I do no understand what Cohen means by introducing tan(-x) = - tan x.

Have you learned (or looked up) the concept of odd functions and symmetry about the origin?

When f(-x) = -f(x), for all x, that means that whenever (x, y) is a point on the graph of f, then so is (-x, -y). These points are on opposite sides of the origin, so we call this origin symmetry.

Visually, you can imagine putting a pin in the graph at the origin and spinning the graph 180 degrees, and it will look identical. So this symmetry makes it a little easier to draw the graph: just draw the right half, spin it around, and you have the left half.

He then goes on to say the following which makes no sense to me. In this picture, Cohen introduces the unit circle. I do not see the connection between the unit circle and the tangent function as used here.

I think it is easier for me just to watch video clips on You Tube showing how to graph the trig functions. At the same time, I would like to understand my textbooks. Otherwise, it makes no sense to have math textbooks in my small library of books.

In the picture, points P and Q are opposite; as indicated by the arrows, they are pi radians (180 degrees) apart. The tangents of these two angles are y/x for P, and (-y)/(-x) = y/x for Q, so they are equal. (On the unit circle diagram, the tangent of an angle is just the slope of the line.) That is, angles 180 degrees apart always have the same tangent, so the period of the tangent function is pi.

It may be that you need to see moving images, not just the static form in a book, even though authors try as hard as they can to make their still images understandable. In a class, I would be pointing at the graph as I talked about it, and maybe using an animation to make it more visible.

 

[harpazo]

Yes, it has asymptotes x = -pi/2 and x = +pi/2, as seen in the graph. There are more at x = 3pi/2, 5pi/2, and so on; in general, any odd multiple of pi/2 (that is, ..., -5, -3, -1, 1, 3, 5, ... times pi/2). The distance from any of these to the next is pi, the period of the tangent function.


Have you learned (or looked up) the concept of odd functions and symmetry about the origin?

When f(-x) = -f(x), for all x, that means that whenever (x, y) is a point on the graph of f, then so is (-x, -y). These points are on opposite sides of the origin, so we call this origin symmetry.

Visually, you can imagine putting a pin in the graph at the origin and spinning the graph 180 degrees, and it will look identical. So this symmetry makes it a little easier to draw the graph: just draw the right half, spin it around, and you have the left half.


In the picture, points P and Q are opposite; as indicated by the arrows, they are pi radians (180 degrees) apart. The tangents of these two angles are y/x for P, and (-y)/(-x) = y/x for Q, so they are equal. (On the unit circle diagram, the tangent of an angle is just the slope of the line.) That is, angles 180 degrees apart always have the same tangent, so the period of the tangent function is pi.

It may be that you need to see moving images, not just the static form in a book, even though authors try as hard as they can to make their still images understandable. In a class, I would be pointing at the graph as I talked about it, and maybe using an animation to make it more visible.

You asked:

"Have you learned (or looked up) the concept of odd functions and symmetry about the origin?"

Yes, I slightly recall what Sullivan said about this concept in my College Algebra textbook Edition 9. It has been almost two years since studying this origin symmetry idea with Sullivan (another favorite author). Tell me, in light of COVID-19, are you teaching math online?

 

[Dr.Peterson]

Lately I've been teaching only in the fall semesters (when there is need for more faculty), so I missed the opportunity to teach remotely in the spring. I was tutoring in the learning center, which became an experiment in remote tutoring. It didn't work well because not enough students tried it, and even fewer got past the technical hurdles; it worked well for some who did. We'll be trying a different way for the summer, possibly leading into the fall. Over the summer I have to learn best practices for remote teaching in case we have to in the fall, learning from what worked and what didn't for others. (We just had a Zoom meeting on that topic.)

 

[harpazo]

Lately I've been teaching only in the fall semesters (when there is need for more faculty), so I missed the opportunity to teach remotely in the spring. I was tutoring in the learning center, which became an experiment in remote tutoring. It didn't work well because not enough students tried it, and even fewer got past the technical hurdles; it worked well for some who did. We'll be trying a different way for the summer, possibly leading into the fall. Over the summer I have to learn best practices for remote teaching in case we have to in the fall, learning from what worked and what didn't for others. (We just had a Zoom meeting on that topic.)

It is my hope that a vaccine to control COVID-19 is developed before the Fall.