How can arctan(x) be expressed using logs?

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Discussion Overview

The discussion revolves around expressing the arctangent function in terms of logarithms, exploring various mathematical approaches and methods. The scope includes theoretical reasoning, mathematical reasoning, and potential applications of different mathematical techniques.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant mentions that their math teacher indicated arctan can be expressed using logarithms and seeks clarification on how to achieve this.
  • Another participant suggests solving the equation tan(y) = x and notes the prerequisite of expressing tan(y) in terms of exponentials.
  • A different participant proposes using Euler's formula to express arctan, presenting a formula involving complex exponentials.
  • One participant confirms the use of Euler's formula and indicates that solving a quadratic equation will be necessary in the process.
  • A follow-up question asks for clarification on what specific quadratic equation is involved in the solution.
  • Another participant provides a transformation of the tangent function into a form that suggests a quadratic equation will arise from manipulating the equation.
  • A participant expresses understanding of the previous explanations.

Areas of Agreement / Disagreement

Participants appear to be exploring different methods to express arctan using logarithms, with no consensus reached on a single approach or solution. Multiple competing views and techniques remain present in the discussion.

Contextual Notes

The discussion involves assumptions about knowledge of exponential functions and transformations, as well as the potential complexity of the methods mentioned, which may not be fully resolved in the posts.

Who May Find This Useful

Readers interested in mathematical expressions of trigonometric functions, particularly those exploring connections between trigonometry and logarithmic functions, may find this discussion relevant.

cragar
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my math teacher said that the arctan can be setup in terms of logs
does anyone know how to do this.
 
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Solve tan(y)=x. (Prerequisite: you must know how to write tan(y) in terms of exponentials)

Other methods are possible (e.g. antidifferentiate f(x)=1/(1+x²)), but there are more technical details involved.
 
can i use eulers formula to do it .

so would it be [(e^(ix)-e^(-ix)]/[(ie^(ix)+ie^(-ix))] = tan(x)
 
Last edited:
Correct.

You'll need to solve a quadratic in the process.
 
what quadratic
 
You have
[itex]tan x= \frac{e^x- e^{-x}}{e^x+ e^{-x}}= y[/itex]
First multiply on both sides of the equation by [itex]e^x+ e^{-x}[/itex].
Then multiply both sides o the equation by [itex]e^x[/itex]
 
oh i see
 

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