Trigonometric identity for inverse tan

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Homework Help Overview

The discussion revolves around the trigonometric identity involving the inverse tangent function, specifically exploring the relationship between tan^(-1)(x) and tan^(-1)(1/x). Participants are examining how this identity may extend when a constant is introduced into the argument of the function.

Discussion Character

  • Conceptual clarification, Assumption checking, Exploratory

Approaches and Questions Raised

  • Participants are questioning whether the identity holds when substituting variables, particularly when replacing x with y or introducing a constant a. There is also exploration of whether the constant should be placed in the numerator or denominator in the context of the identity.

Discussion Status

Some participants have suggested using geometric interpretations, such as drawing a right triangle, to validate the identity. There is a mix of opinions on the validity of the identity under different substitutions, with some participants expressing confidence in their conclusions while others seek confirmation.

Contextual Notes

Participants are working within the constraints of a homework help context, where definitive solutions are not provided, and assumptions about variable substitutions are being actively discussed.

Kat007
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Hello,

Could you please clarify if this is correct:

If tan^(-1)(x) = Pi/2 - tan^(-1)(1/x)

Then if we have (ax) as the angle where a is a constant, do we get:
tan^(-1)(ax) = Pi/2 - tan^(-1)(a/x)
or does the constant go on the bottom with the x? i.e. or:
tan^(-1)(ax) = Pi/2 - tan^(-1)(1/(ax))

Thank you very much,
Kat
 
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Kat007 said:
Hello,

Could you please clarify if this is correct:

If tan^(-1)(x) = Pi/2 - tan^(-1)(1/x)

Then if we have (ax) as the angle where a is a constant, do we get:
tan^(-1)(ax) = Pi/2 - tan^(-1)(a/x)
or does the constant go on the bottom with the x? i.e. or:
tan^(-1)(ax) = Pi/2 - tan^(-1)(1/(ax))

Thank you very much,
Kat

I believe that you can answer this yourself by drawing a right triangle (use a 30-60-30 triangle), and writing the tangent in terms of the ratio of the opposite and adjacent sides. Then double the height of the opposite side...
 
Do you think the identity should change if you replace the letter x with the letter y? Is the identity invalid for some value of y? What if y = ax? If you agree that it is still valid, what do you get when you put ax in the place of y?
 
OK so it seems to work out as the
tan^(-1)(ax) = Pi/2 - tan^(-1)(1/(ax))
Could you please confirm that this is correct?
 
Kat007 said:
OK so it seems to work out as the
tan^(-1)(ax) = Pi/2 - tan^(-1)(1/(ax))
Could you please confirm that this is correct?

Which method did you use to conclude that? Giga and I have a bet going...
 
lol how kind of you.. Jokers heheee
The triangle one.
They both makes sence. Thanks,
Kat
 
Kat007 said:
lol how kind of you.. Jokers heheee
The triangle one.
They both makes sence. Thanks,
Kat

Oh fooey. If they both make sense, then neither of us wins the bet. Oh well :smile:
 
Nah, Giga's make more sense AND no pen and paper required :-p
 

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