Euler's Formula of Inverse Trig Functions

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Discussion Overview

The discussion centers around the existence of a clean exponential formula for inverse trigonometric functions, specifically arcsine and arccosine, similar to the formulas for sine and cosine. The scope includes theoretical exploration and mathematical reasoning.

Discussion Character

  • Exploratory, Mathematical reasoning

Main Points Raised

  • Some participants assert that there is no clean exponential formula for inverse trigonometric functions, expressing this with humor and frustration.
  • One participant suggests that while a clean formula may not exist, arcsine and arccosine can be expressed using logarithms and square roots, proposing a method involving the quadratic equation derived from the sine function.

Areas of Agreement / Disagreement

Participants generally agree that a clean exponential formula does not exist, but there is a divergence in the methods proposed for expressing inverse trigonometric functions.

Contextual Notes

The discussion does not resolve the limitations of the proposed logarithmic and square root expressions, nor does it clarify the assumptions underlying the quadratic approach mentioned.

Tac-Tics
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Is there a clean exponential formula for inverse trig functions (arcsine, arccosine) like there is for sine and cosine?
 
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Tac-Tics said:
Is there a clean exponential formula for inverse trig functions (arcsine, arccosine) like there is for sine and cosine?

Short answer … no! :redface:

Long answer … noooooooo! :cry:

You knew that didn't you? :smile:​
 
tiny-tim said:
Short answer … no! :redface:

Long answer … noooooooo! :cry:

You knew that didn't you? :smile:​

I figured as much. Mathematics is as cruel as she is beautiful.
 
You can write them with logarithms and square roots: first write the formula for, say, sin(z) in terms of exp(iz) and exp(-iz). Then solve for exp(iz) (it's a quadratic equation). Then take logarithms.
 

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