The graph of sin inverse (sin x) after the domain of (- pi/2, pi/2)

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Discussion Overview

The discussion revolves around the graph of the function sin inverse(sin x) after the domain of (-π/2, π/2). Participants explore the behavior of this function outside its principal value range and seek to understand the logical derivation of its expressions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that within the domain of (-π/2, π/2), the graph of sin inverse(sin x) is y = x, while others challenge this by noting that the graphs are not identical outside this interval.
  • There is a suggestion that the many-to-one nature of the sine function affects the behavior of sin inverse(sin x) after crossing the principal value domain.
  • One participant mentions the need for logical derivation of the expressions and asks for clarification on how the graph behaves outside the principal value range.
  • Another participant describes the behavior of the function in the interval [π/2, 3π/2], indicating that it transitions to a different linear equation with a slope of -1.
  • There is a reference to the analysis required for each interval of the function, suggesting that the overall graph resembles a sawtooth pattern.

Areas of Agreement / Disagreement

Participants generally agree that the graph of sin inverse(sin x) behaves as y = x within the interval (-π/2, π/2), but there is disagreement regarding the nature of the graph outside this interval, leading to multiple competing views on its behavior.

Contextual Notes

Some participants express uncertainty about the derivation of the expressions and the intuitive understanding of the graph's behavior, indicating that further clarification may be needed.

esha
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the graph of sin inverse (sin x) between the domain of ( -pi/2,pi/2) is y = x. but after it crosses that domain of course the expression won't be the same anymore because sin inverse has its principle value as ( - pi/2, pi/2) due to sin x many to one natured function. now the way these expressions change is what doesn't seem intuitive to me. can anybody please tell me how to derive those expressions logically?
 
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esha said:
the graph of sin inverse (sin x) between the domain of ( -pi/2,pi/2) is y = x.
No, it isn't. The graphs of the two functions are close together when x is in the interval [-.5, .5], but they aren't identical.
esha said:
but after it crosses that domain of course the expression won't be the same anymore because sin inverse has its principle value as ( - pi/2, pi/2) due to sin x many to one natured function. now the way these expressions change is what doesn't seem intuitive to me. can anybody please tell me how to derive those expressions logically?
What expression? Are you trying to understand why the graph of ##y = \sin^{-1}(x)## looks the way it looks? Are you asking how the values on this graph are calculated? If you have studied calculus, one of the topics presented later is infinite series. One such series is the expansion for the arcsine function. See https://math.stackexchange.com/questions/197874/maclaurin-expansion-of-arcsin-x.
 
Last edited:
its not sin inverse graph..
that's pretty straight forward..
. m talking bout sin inverse ( sin x) graph
 
they are different...
 
esha said:
its not sin inverse graph..
that's pretty straight forward..
. m talking bout sin inverse ( sin x) graph
It wasn't clear to me what you were asking about, which is the graph of ##y = \sin^{-1}(\sin(x))##. On the interval ##[-\pi/2, \pi/2]##, the graph of this function is the same as that of y = x, which is what you said.

On the interval ##[\pi/2, 3\pi/2]##, the graph of ##y = \sin^{-1}(\sin(x))## goes from ##(\pi/2, \pi/2)## down to (0, 0), and then to ##(3\pi/2, -\pi/2)##, so this line segment has a slope of -1 with an equation of ##y = -(x - \pi)## (in other words, the same as the graph of y = -x, but shifted to the right by ##\pi## units.

You have to go through this kind of analysis on each of the intervals ##[-\pi/2 + 2n\pi, \pi/2 + 2n\pi]##. Doing this you get a sawtooth graph like this one:http://www.wolframalpha.com/input/?i=plot+y+=+arcsin(sin(x))
 
thanks... for the explanation
 

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