Asymptotes of tan (x)

[harpazo]

According to the texbook, the graphs of y = tan x, cot x, csc x and sec x all possess asymptotes. Cohen goes on to say that for the trig identity tan x = sin x/cos x, at x = pi/2, the denominator is 0. Furthermore, when x is equal to any ODD INTEGRAL MULTIPLE of pi, the denominator of tan x = sin x/cos x will be 0 and, as a result, tan x will be undefined.

Let x = pi/2

tan (pi/2) = sin(pi/3)/cos(pi/2)

tan(pi/2) = [sqrt{3}/2]/0

tan(pi/2) = 0 = undefined

1. I understand ASYMPTOTE to mean that 0 is not allowed in the denominator of a fraction. If this is untrue, what does ASYMPTOTE mean for us in terms of tan x?

2. In what way is the idea of asymptote related to negative and positive infinity?

3. Cohen stated: "We'll obtain the graph of the tangent function by a combination of both point-plotting and symmetry considerations." What does he mean by SYMMETRY CONSIDERATION?

4. When x is equal to any ODD INTEGRAL MULTIPLE of pi, the denominator of tan x = sin x/cos x will be 0 and, as a result, tan x will be undefined. What is an ODD INTEGRAL MULTIPLE OF pi? Can I have a list, say, of 5 odd integral multiples of pi?

 

[Harry_the_cat]

1. An asymptote is a line that is approached but never crossed. Eg, look at the grraph of y=1/x. The x and y axes are both asymptotes. If y= tan(x), y= pi/2 is one asymptote.
2. Look at graph of y= tan(x). As x approaches pi/2 from the left, what does y do? It goes off to infinity.
3. Probably properties like tan(-x)= - tan(x).
4. It should say an odd integral multiple of pi/2. An odd integer is 1, 3, 5... also -1, -3, -5 etc. So an odd integral multiple of pi includes pi, 3pi, 5pi, -1 pi, -3pi, -5pi etc.

 

[Otis]

… What does he mean by SYMMETRY CONSIDERATION?

If I gave you some points on the right-half of a parabola, you could graph the left-half, too, because of symmetry (the left half is a reflection of the right half, and vice versa). A parabola is symmetrical about its central axis, so you'd reflect the given points across that axis, to graph the other half.

Trig functions are periodic; their graphs repeat the same shape. Once you have the curve in one interval, you can copy it into others.

?

 

[Otis]

… An asymptote is a line that is approached but never crossed …

Mostly, but not always.

 

[harpazo]

Mostly, but not always.

Nice link about graphs of functions.

 

[harpazo]

1. An asymptote is a line that is approached but never crossed. Eg, look at the grraph of y=1/x. The x and y axes are both asymptotes. If y= tan(x), y= pi/2 is one asymptote.
2. Look at graph of y= tan(x). As x approaches pi/2 from the left, what does y do? It goes off to infinity.
3. Probably properties like tan(-x)= - tan(x).
4. It should say an odd integral multiple of pi/2. An odd integer is 1, 3, 5... also -1, -3, -5 etc. So an odd integral multiple of pi includes pi, 3pi, 5pi, -1 pi, -3pi, -5pi etc.

1. I did not know that negative numbers can be considered odd integers or odd multiples.

2. You said:

"So an odd integral multiple of pi includes pi, 3pi, 5pi, -1 pi, -3pi, -5pi etc."

Why is - 1 on the list? The integer - 1 is listed without pi next to it. What would be an odd multiple of 7pi?

 

[harpazo]

If I gave you some points on the right-half of a parabola, you could graph the left-half, too, because of symmetry (the left half is a reflection of the right half, and vice versa). A parabola is symmetrical about its central axis, so you'd reflect the given points across that axis, to graph the other half.

Trig functions are periodic; their graphs repeat the same shape. Once you have the curve in one interval, you can copy it into others.

?

I link symmetry to mirror-like images.

 

[Dr.Peterson]

There are several kinds of symmetry, not only reflection symmetry which is what we first think of.

They are all worth being familiar with:

Symmetry - Reflection and Rotation

Learn about the different types of symmetry: Reflection Symmetry (sometimes called Line Symmetry or Mirror Symmetry), Rotational Symmetry and Point Symmetry.

www.mathsisfun.com

Symmetry in Geometry - MathBitsNotebook(Geo)

Those are about geometry, but it transfers directly to algebra:

Symmetry in Equations

Equations can have symmetry ... In other words, there is a mirror-image. ... The benefits of finding symmetry in an equation are

www.mathsisfun.com

Symmetry and Graphs | Purplemath

Demonstrates how to recognize symmetry in graphs, in particular with respect to the y-axis and the origin.

www.purplemath.com

Then you can get into three dimensions ...

 

[harpazo]

There are several kinds of symmetry, not only reflection symmetry which is what we first think of.

They are all worth being familiar with:

Symmetry - Reflection and Rotation

Learn about the different types of symmetry: Reflection Symmetry (sometimes called Line Symmetry or Mirror Symmetry), Rotational Symmetry and Point Symmetry.

www.mathsisfun.com

Symmetry in Geometry - MathBitsNotebook(Geo)

Those are about geometry, but it transfers directly to algebra:

Symmetry in Equations

Equations can have symmetry ... In other words, there is a mirror-image. ... The benefits of finding symmetry in an equation are

www.mathsisfun.com

Symmetry and Graphs | Purplemath

Demonstrates how to recognize symmetry in graphs, in particular with respect to the y-axis and the origin.

www.purplemath.com

Then you can get into three dimensions ...

Wow! Great reply! Thank you for the links.

 

[Harry_the_cat]

1. I did not know that negative numbers can be considered odd integers or odd multiples.

2. You said:

"So an odd integral multiple of pi includes pi, 3pi, 5pi, -1 pi, -3pi, -5pi etc."

Why is - 1 on the list? The integer - 1 is listed without pi next to it. What would be an odd multiple of 7pi?

It's -1pi. There's no comma after -1.

 

[Harry_the_cat]

I link symmetry to mirror-like images.

There is rotational symmetry as well as reflective symmetry.

 

[harpazo]

It's -1pi. There's no comma after -1.

Ok....

 

[harpazo]

There is rotational symmetry as well as reflective symmetry.

I plan to study and review geometry, at least at the high school level in the future. I never want to stop learning and improving my math skills.