Tangent Line and Coordinates of Trigonometric Function

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Homework Help Overview

The discussion revolves around finding the x- and y-coordinates of points on the curve defined by the function y = sin(x) / (√2 - cos(x)) where the tangent line is horizontal. The subject area includes calculus, specifically the application of derivatives to analyze the behavior of functions.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to apply the quotient rule to find the derivative of the function but expresses difficulty in progressing towards a solution. Some participants question the necessity of rationalizing √2 and clarify its role as a constant. Others request the original poster to share their work on the derivative to facilitate further discussion.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the derivative and its implications for finding horizontal tangents. There is no explicit consensus yet, but some guidance has been offered regarding the treatment of constants in differentiation.

Contextual Notes

Participants are navigating the complexities of applying the quotient rule and addressing potential misunderstandings about constants in the context of derivatives. The original poster has not yet provided their detailed work, which may be necessary for further assistance.

gabyoh23
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Homework Statement



There are infinitely many points on the curve y = [tex]\frac{sin x}{\sqrt{2}- cos x}[/tex] at which the tangent line to this curve is horizontal. Find the x- and y-coordinates of one such point.

Homework Equations



y' = slope of the tangent line
Etc., etc.

The Attempt at a Solution


I know you have to take the derivative of the given equation, and at first, I tried using the quotient rule, but I got nowhere with that. Then I tried rationalizing the [tex]\sqrt{2}[/tex], but that didn't really get me anywhere either. I also have no idea how to find the x- and y-coordinates.

All help is greatly appreciated!
 
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Nothing needs to be done with [tex]\sqrt{2}[/tex], it is a constant. What's the derivative of a constant? (btw, [tex]\sqrt{2}[/tex] is an irrational number which is why you couldn't rationalize it.)

The coordinates are the x and y-values of the function and y=f(x) so, (x,y)=(x,f(x))
 
To the OP, please show us your work in taking the derivative using the quotient rule. I did the same and got a particularly pleasing answer.
 
I got something like [tex]\frac{1-cos^2(x)}{2-cos^2(x)}[/tex].
I'm not sure if this right at all. My prowess with the quotient rule is shoddy at best.
 

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