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

In summary, the conversation discussed the properties, identities and usages of composite trigonometric functions, particularly sin(sinx) and cos(sin(x)). It was mentioned that the argument of a goniometric function is usually an angle, but composite functions can be applied in substitutions. There was also a question about the difference between goniometric functions and trigonometric functions, to which it was stated that they are essentially the same thing.
  • #1
ddddd28
73
4
Hello,
Are there any particular properties, indentities or usages of composite trigonometric functions, say sin(sinx) or cos(sin(x))?
 
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  • #2
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:
  • #3
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?
 
  • #4
This is a combination of a goniometric function acting on the result of an inverse goniometric function. Makes sense.
 
  • #5
OK. By the way, is there any difference between goniometric functions and trigonometric functions? or they are just two names of the same thing?
 
  • #6
Same thing
 
  • #7
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...
 

1. What is a trigonometric function composition?

A trigonometric function composition is an expression in which one trigonometric function is used as the input of another trigonometric function. For example, sin(sin(x)) is a trigonometric function composition.

2. What are the characteristics of trigonometric function compositions?

Trigonometric function compositions have the following characteristics:

  • The outer function operates on the result of the inner function.
  • The inner function can be any trigonometric function or a combination of trigonometric functions.
  • The composition can be simplified using trigonometric identities.
  • The period of the composition may be different from the individual functions.
  • Trigonometric function compositions can have multiple solutions.

3. How do you simplify a trigonometric function composition?

To simplify a trigonometric function composition, you can use trigonometric identities such as the double angle or half angle identities. You can also use the sum and difference identities to rewrite the composition in a simpler form.

4. Can trigonometric function compositions have infinite solutions?

Yes, trigonometric function compositions can have infinite solutions. This is because trigonometric functions are periodic, meaning they repeat themselves at regular intervals. Therefore, the composition of two or more trigonometric functions can produce multiple solutions that satisfy the equation.

5. How are trigonometric function compositions used in real life?

Trigonometric function compositions have various applications in fields such as physics, engineering, and astronomy. For example, in physics, trigonometric function compositions are used to model the motion of objects in circular or oscillating motion. In engineering, they are used to design and analyze structures, such as bridges and buildings. In astronomy, they are used to calculate the positions and movements of celestial objects.

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