What is the attempt at a solution for substituting pi/14?

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Homework Help Overview

The discussion revolves around the substitution of \(\frac{\pi}{14}\) in a trigonometric context, specifically focusing on the behavior of the tangent function as the variable approaches this value. Participants explore the implications of this substitution on the tangent function and its exponential counterpart.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the behavior of \(\tan(7x)\) as \(x\) approaches \(\frac{\pi}{14}\) from above, questioning the values obtained through substitution and the implications of using different approximations of \(\pi\).

Discussion Status

The discussion is active, with participants sharing calculations and questioning the accuracy of their methods. There is recognition of the behavior of \(\tan(7x)\) approaching negative infinity as \(x\) approaches \(\frac{\pi}{14}\) from the right, and some participants express agreement on this observation.

Contextual Notes

Some participants note the importance of calculator settings, particularly whether they are in degrees or radians, which affects the results of their calculations. There is also a focus on the limits and behavior of functions near critical points.

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Homework Statement



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Homework Equations





The Attempt at a Solution



I tried substituting pi/14 it doesn't accept that answer.
 
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7*pi/14=pi/2. What's the behavior of tan(x) like as x->pi/2 from above?
 


it gets close to 1.57~
 


7*x gets close to pi/2=1.57~, as x->(pi/14)+, sure. But x->(pi/14)+ means thinking of x's a little larger than pi/14. What's tan(7*(pi/14+0.01)), tan(7*(pi/14+0.001)) etc etc. Punch them into a calculator if you have to.
 


tan(7 ( pi / 14 +.01 ) ) is .028645
tan(7 ( pi / 14 +.001 ) ) is .027545
tan(7 ( pi / 14 +.0001 ) ) is .027435
tan(7 ( pi / 14 +.00001 ) ) is .027424

so basically, its approaching .02742
 


I get tan(7*((pi/14)+0.001))=-1428.57... Are you using an bad approximation of pi? Don't. 7*pi/14+(a small number) is pi/2+(a small number). What's tan(pi/2+(a small number))? Look at a graph of tan.
 


I got -999.9996667 for tan(pi/2+.001)

my calculator was in degrees. ;O
 


eplymale3043 said:
I got -999.9996667 for tan(pi/2+.001)

my calculator was in degrees. ;O

I forgot about that way of making a mistake. Good for you for catching it. Now if 0.001 gets even smaller??
 


.0001 ~-9999.999967

but when I use .00001 my calculator rounds to -100000 but it should be around -99999.9999967

right?
 
  • #10


Sure, sure. But the point is that tan(7x) is going to minus infinity as x->(pi/14)+, right? Again look at a graph of tan(x) just above pi/2. Agreed?
 
  • #11


Yes, I see it is approaching minus infinity, I can also see it on the graph as well.
 
  • #12


Good, good. So what does e^(tan(7*x)) approach?
 
  • #13


negative infinity?
 
  • #14


tan(7x) approaches negative infinity. What's, oh, say e^(-100000000)? If your calculator overflows, try e^(-100).
 
  • #15


0.

as the number decreases, x approaches 0.

so the limit would be 0?
 
  • #16


You said it, not me. But x doesn't approach 0. e^tan(7*x) approaches 0 as x->(pi/14)+. Sure, 0.
 

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