Characteristics of trigonometric function compositions like sin(sin(x))

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Discussion Overview

The discussion revolves around the properties, identities, and applications of composite trigonometric functions, specifically examples like sin(sin(x)) and cos(sin(x)). The scope includes theoretical considerations and potential applications in mathematical substitutions.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • Some participants suggest that composite trigonometric functions may not have many established properties, as the argument is typically an angle resulting in a numerical output.
  • One participant mentions finding online that composite trigonometric functions can be useful in substitutions, providing an example of replacing √1-9x^2 with sin(arccos 3x).
  • Another participant notes that this involves a goniometric function acting on the result of an inverse goniometric function, indicating a logical connection.
  • A question is raised regarding the difference between goniometric and trigonometric functions, with some asserting they are the same.
  • One participant elaborates that while a sine function is a ratio of lengths, an angle can also be viewed as a ratio of arc length to radius, suggesting a deeper conceptual understanding.

Areas of Agreement / Disagreement

Participants express varying views on the properties and applications of composite trigonometric functions, with no clear consensus on their usefulness or distinct characteristics.

Contextual Notes

Some limitations include the lack of established identities for composite functions and the ambiguity surrounding the terminology of goniometric versus trigonometric functions.

ddddd28
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Hello,
Are there any particular properties, indentities or usages of composite trigonometric functions, say sin(sinx) or cos(sin(x))?
 
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Hi d5,

Not many I should think. Usually the argument of a goniometric function is an angle and the result is a number, not an angle.
But it's always nice to have students sink their teeth in such a contraption.
 
Last edited:
Well, I have just found on the web that composite trigonometric functions can be applied in substitutions, for example: √1-9x^2 can be replaced to sin(arccos 3x).Is it useful somehow?
 
This is a combination of a goniometric function acting on the result of an inverse goniometric function. Makes sense.
 
OK. By the way, is there any difference between goniometric functions and trigonometric functions? or they are just two names of the same thing?
 
Same thing
 
BvU said:
Usually the argument of a goniometric function is an angle and the result is a number, not an angle.
But it's always nice to have students sink their teeth in such a contraption.
Not completely fair: a sine is the ratio of the lengths of two sides. One can consider an angle as the ratio of arc length to radius...
 

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