- #1

- 23

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## Homework Statement:

- Tan ##2 \theta=4 /(1-1)##. This means ##2 \theta=90^{\circ}## and ##\theta=45^{\circ}##

## Relevant Equations:

- None

One of my solutions had this in one part. Why is this the case?

- Thread starter ElectronicTeaCup
- Start date

- #1

- 23

- 1

## Homework Statement:

- Tan ##2 \theta=4 /(1-1)##. This means ##2 \theta=90^{\circ}## and ##\theta=45^{\circ}##

## Relevant Equations:

- None

One of my solutions had this in one part. Why is this the case?

- #2

etotheipi

2020 Award

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For what values of ##\alpha## does ##\tan{\alpha}## diverge to positive infinity?

- #3

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You know that as ##\theta \rightarrow \frac \pi 2## then ##\tan \theta \rightarrow +\infty##?Homework Statement::Tan ##2 \theta=4 /(1-1)##. This means ##2 \theta=90^{\circ}## and ##\theta=45^{\circ}##

Relevant Equations::None

One of my solutions had this in one part. Why is this the case?

- #4

- 23

- 1

Also, is this also correct?

##\begin{array}{l}

\cot 2 \theta=0 \\

\frac{\cos 2 \theta}{\sin 2 \theta}=0 \\

\cos 2 \theta=0 \\

2 \theta=90

\end{array}##

- #5

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That's right where ##2\theta## is between ##0## and ##180°##. Another solution could be ##2\theta = 270°##.

Also, is this also correct?

##\begin{array}{l}

\cot 2 \theta=0 \\

\frac{\cos 2 \theta}{\sin 2 \theta}=0 \\

\cos 2 \theta=0 \\

2 \theta=90

\end{array}##

- #6

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Isn't ##1-1## a bit ambiguous, though? Was this a limit like ##x-1##?

- #7

etotheipi

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A related example... if we parameterise a circle with ##(x,y) = (a\cos{\theta}, a\sin{\theta})## s.t. ##y' = -\frac{\cos{\theta}}{\sin{\theta}}## and wanted to see where the circle is vertical, would you also take issue with identifying ##\sin{\theta} = 0 \implies \theta = 0, \pi##?Isn't ##1-1## a bit ambiguous, though? Was this a limit like ##x-1##?

- #8

Mark44

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1 - 1 = 0, which is not at all ambiguous. However, the fraction ##\frac 4 {1 - 1}## is undefined.Isn't ##1-1## a bit ambiguous, though? Was this a limit like ##x-1##?

- #9

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No problem with identifying the vertical part, but ##y'## either goes to infinity or negative infinity depending on how you choose to make ##\theta## go to either of those values (right or left of ##0## or ##\pi##).A related example... if we parameterise a circle with ##(x,y) = (a\cos{\theta}, a\sin{\theta})## s.t. ##y' = -\frac{\cos{\theta}}{\sin{\theta}}## and wanted to see where the circle is vertical, would you also take issue with identifying ##\sin{\theta} = 0 \implies \theta = 0, \pi##?

The same thing with the ##1/(1-1)## fraction. Whether you have it ##1-x## or ##x-1##, then, however ##x## approaches ##1##, you will get different results (##\pi/2## or ##-\pi/2##).

@Mark44 this is what I was aiming at.

- #10

etotheipi

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I see your point, and I the interpretation would be clearer from context.No problem with identifying the vertical part, but ##y'## either goes to infinity or negative infinity depending on how you choose to make ##\theta## go to either of those values (right or left of ##0## or ##\pi##).

The same thing with the ##1/(1-1)## fraction. Whether you have it ##1-x## or ##x-1##, then, however ##x## approaches ##1##, you will get different results (##\pi/2## or ##-\pi/2##).

@Mark44 this is what I was aiming at.

In most cases where I've seen these sorts of steps, it hasn't mattered from which side you approach ##x## (e.g. for my circle example, a gradient of ##\infty## is equivalent to one of ##-\infty## for purposes of determining the vertical tangent).

- #11

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$$\tan 2x = \frac{2 \tan x}{1-\tan^2 x} = \frac{4}{0} \rightarrow 0 = 4-4\tan^2 x \rightarrow \tan^2 x = 1$$

And from here the ##\frac{\pi}{4}## falls out "neatly", along with all the other solutions. But be careful, this is "illegal" math. However, you started off with a statement of ##\frac{4}{0}##, I feel okay using hand-wavy tactics here.

- #12

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Is that really the problem as given to you? I suggest that you had done some work to arrive at that. If so, please post the actual problem.Homework Statement::Tan ##2 \theta=4 /(1-1)##.

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