How to simplify an iterated trigonometric expression

In summary, the conversation discusses various ways to simplify the expression cos(sin x). These include using Taylor series, matrix multiplication, and exploring relationships with Bessel functions. Some possible simplifications were suggested, such as using the identity cos^2(x) + sin^2(x) = 1, but it was determined that none of these methods would effectively simplify the given expression.
  • #1
Leo Liu
353
156
Homework Statement
.
Relevant Equations
.
eg ##\cos (\sin x)##
Asking this question out of curiosity.
 
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  • #2
Leo Liu said:
Homework Statement:: .
Relevant Equations:: .

eg ##\cos (\sin x)##
Asking this question out of curiosity.
I don't see how you're going to be able to get anything simpler than that.
 
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  • #3
Typically this can only be done with the partial sums of the Taylor series of the functions. There are a variety of ways to calculate a partial sum of the composition, including matrix multiplication.
 
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  • #4
If you are interested about the Fourier series of cos(cos x) or cos (sin x) I think they are related to the Bessel functions.
 
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  • #5
Delta2 said:
If you are interested about the Fourier series of cos(cos x) or cos (sin x) I think they are related to the Bessel functions.
Thank you this is the best answer I got :D
 
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  • #6
Or maybe you want something like :

##Cos^2(sinx)+ Sin^2(sinx)=1##

So that ##Cos(sinx)=\sqrt {1-Sin^2(sinx)}##?
 
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  • #7
WWGD said:
Or maybe you want something like :

##Cos^2(sinx)+ Sin^2(sinx)=1##

So that ##Cos(sinx)=\sqrt {1-Sin^2(sinx)}##?
I thought of something like this, as well as a Taylor or Maclaurin series, but none of these seemed like they would serve to simplify the given expression.
 
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Related to How to simplify an iterated trigonometric expression

1. What is an iterated trigonometric expression?

An iterated trigonometric expression is a mathematical expression that involves multiple trigonometric functions, such as sine, cosine, and tangent, that are repeated multiple times within the expression.

2. Why is it important to simplify an iterated trigonometric expression?

Simplifying an iterated trigonometric expression can make it easier to understand and work with in further calculations. It can also help to identify patterns and relationships within the expression.

3. What are some common techniques for simplifying an iterated trigonometric expression?

Some common techniques for simplifying an iterated trigonometric expression include using trigonometric identities, factoring, and combining like terms.

4. Can all iterated trigonometric expressions be simplified?

No, not all iterated trigonometric expressions can be simplified. Some may already be in their simplest form or may not have any simplification rules that apply to them.

5. How can I check if my simplified iterated trigonometric expression is correct?

You can check your simplified iterated trigonometric expression by substituting in values for the variables and comparing the result to the original expression. You can also use a calculator or online tool to verify the simplification.

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