Is Sin^-1 X = (Sin X ) ^ -1 ?from my book it stated :f(x) =

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Discussion Overview

The discussion centers around the notation and interpretation of the inverse sine function, specifically whether Sin^-1 X is equivalent to (Sin X)^-1. Participants explore the implications of this notation in calculus and its potential for confusion among learners.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants assert that Sin^-1 X is not equal to (Sin X)^-1, clarifying that Sin^-1 X denotes the arcsine function, which is the inverse of the sine function.
  • One participant emphasizes the importance of understanding the derivative of inverse trigonometric functions, suggesting that this is typically covered in introductory calculus courses.
  • Concerns are raised about the misleading nature of using Sin^-1 for arcsine, particularly for beginners, and the preference for using the notation arcsin instead.
  • Another participant notes that the notation with -1 in the exponent is perpetuated by software tools, which may contribute to confusion in educational contexts.
  • There is a discussion about the general representation of inverse functions, where some participants express that while f^-1 is standard for function inverses, it can lead to misunderstandings when compared to multiplicative inverses.
  • Some participants share personal experiences of explaining the distinction to peers, noting that many are surprised to learn that Sin^-1 X does not equal 1/Sin X.

Areas of Agreement / Disagreement

Participants generally agree that the notation Sin^-1 can be misleading, especially for beginners, but there is no consensus on the best way to represent inverse functions or the implications of this notation in educational materials.

Contextual Notes

The discussion highlights the potential for confusion arising from different notational conventions and the need for clarity in teaching inverse trigonometric functions. There are unresolved questions about the best practices for notation in educational contexts.

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Is Sin^-1 X = (Sin X ) ^ -1 ??from my book ... it stated :f(x) =

Is Sin^-1 X = (Sin X ) ^ -1 ??

from my book ... it stated :
f(x) = Sin ^-1 X
f'(x) = 1/(1-X^2)^1/2

instead of f'(x) = -cot X / sin X

Why is it so ?
 
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First of all, sin^1(x) =/= 1/(sinx). It's a convention to write it as sin^1(x), but it means arcsin of x, the inverse function of sinx.

Do you know how to take the derivative of inverse trigonometric functions? This is typically covered in any freshman calculus book. Try to see if you can't come up with the answer on your own first, we don't want to rob you of the spirit of discovery. ::smile:
 


thanks ! want to make sure between the relationship with sin X and Sin ^-1 X ... =D
 


It's higly misleading esecially to starters not to use <arcsin> for the inverse of sine. But this unfortunate notation with the -1 in the exponent is propagated also because of software such as maple and mathematica and essentially should stay there and not invade calculus textbooks...
 


dextercioby said:
It's higly misleading esecially to starters not to use <arcsin> for the inverse of sine. But this unfortunate notation with the -1 in the exponent is propagated also because of software such as maple and mathematica and essentially should stay there and not invade calculus textbooks...
I agree that inverse trig functions should first be presented as arcsin, arccos, arctan, etc., rather that sin-1, cos-1, tan-1. However, how can we represent the inverse of an arbitrary function f other than with the standard notation f-1?

The problem is that students confuse expressions such as x-1, the multiplicative inverse of x (or 1/x), with f-1, the function inverse of f, which has nothing to do with multiplication or division.

Exponent notation for function inverses is a convenient and compact form that, like all notations, takes some time to learn.
 


agree with that ... when i tell some of my frens that Sin^-1 X =/= 1/Sin X , they feel shock too ...
 

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