Divergence of Sinx and Cosx: An Explanation

  • Thread starter rcmango
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In summary, the conversation discusses the concept of divergence in trigonometric functions, specifically sin(x) and cos(x). It is explained that these functions do not converge to a specific value as x gets large, but rather their values are bounded. The reason for this is due to the fact that they are periodic and constantly oscillate, making it impossible to "hone in" on a particular value. It is also clarified that divergence does not necessarily mean that the function goes to plus or minus infinity, but rather that it does not converge. The conversation also touches on the concept of superior and inferior limits in these functions.
  • #1
rcmango
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Homework Statement



why does cos x diverge?

Homework Equations





The Attempt at a Solution



is it because it never stops continuing to infinity? it just oscilates until 1?

and does sinx also diverge?

thanks
 
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  • #2
you mean they do not converge as x goes to +/- infinity?
 
  • #3
yes, do they both diverge? as to, going to infinity and never stopping.

also, is this because they have an upper and lower bound, for both, as in -1 and 1 correct?
 
  • #4
both functions do not converge to any particular value as x gets large. their values are bounded but they do not converge
 
  • #5
rcmango said:

Homework Statement



why does cos x diverge?

Homework Equations





The Attempt at a Solution



is it because it never stops continuing to infinity? it just oscilates until 1?

and does sinx also diverge?

thanks

rcmango said:
yes, do they both diverge? as to, going to infinity and never stopping.

also, is this because they have an upper and lower bound, for both, as in -1 and 1 correct?

Do you understand why your first question made no sense? You can take the limit of a function at any value of x. You cannot talk about a limit of a function without specifying where the limit is to be taken. It is trivial that sin(x) and cox(x) converge as, say, x goes to 0 or, for that matter to any real number. Yes, both sin(x) and cos(x) diverge as x goes to infinity or -infinity. It is not because they "both have upper and lower bound". [itex]x^2/(x^2+1)[/itex] has "upper and lower bounds" but its limits, as x goes to either infinity or -infinity, is 0.

Do you understand what "diverge" means? It is not necessary that the value of the function go to plus or minus infinity- diverge simply means that it does not converge- that it does not, here, as x goes to infinity, get closer and closer to some specific number. Yes, both and sin(x) and cos(x) diverge (as x goes to infinity).
 
  • #6
Alot of periodic functions have the same property (im not saying all, ie f(x) = 2 is periodic for any finite interval of choice =] ) but the main reason they do not have a limit is because the value never really "hones" into a particular value, just keeps on changing. Sin x and cos x does have superior and inferior limits though (somewhat obviously) =]
 
  • #7
thankyou, trying to understand series.
 

1. What does it mean for Sinx and Cosx to diverge?

When we say that Sinx and Cosx diverge, it means that the values of these trigonometric functions approach infinity or negative infinity as the input (x) approaches a certain value or as x increases or decreases without bound.

2. Why do Sinx and Cosx diverge?

Sinx and Cosx diverge because they are not defined for certain values of x. For example, the value of Sinx is undefined for x = (2n + 1)π/2 where n is any integer, and the value of Cosx is undefined for x = nπ where n is any integer. This leads to an infinite or undefined output, causing the functions to diverge.

3. How can we determine when Sinx and Cosx will diverge?

We can determine when Sinx and Cosx will diverge by looking at the values of x that make the functions undefined, as mentioned in the previous answer. These values are known as the vertical asymptotes of the functions and indicate where the functions will diverge.

4. Is there a difference between Sinx and Cosx diverging?

Yes, there is a difference between Sinx and Cosx diverging. While Sinx diverges to both positive and negative infinity as x approaches a vertical asymptote, Cosx only diverges to positive or negative infinity depending on the quadrant of the vertical asymptote. For example, if the vertical asymptote is in the first or fourth quadrant, Cosx will diverge to positive infinity, but if it is in the second or third quadrant, Cosx will diverge to negative infinity.

5. How does knowing when Sinx and Cosx diverge affect their graph?

Knowing when Sinx and Cosx diverge can help us accurately graph these functions. The vertical asymptotes indicate where the functions will approach infinity or negative infinity, and this can help us determine the shape and direction of the graph. Without this knowledge, the graph may not be accurate and may not show the complete behavior of the functions.

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