Geometrical interpretation of Taylor series for sine and cosine?

In summary, the speaker has come up with a geometrical interpretation of Taylor's series for sine and cosine. This method involves using purely geometrical constructs instead of summing infinite derivatives over factorials. The speaker is wondering if this has been done before and is hesitant to continue in case it has already been discovered. Initial results show promise, but the complexity increases exponentially with higher levels of accuracy. The speaker is preparing papers to share their findings. The other person is not aware of anything similar, but suggests looking into visual proofs or "proof without words" on the topic.
  • #1
waht
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I've stumbled upon what might be a geometrical interpretation of Taylor's series for sine and cosine. Instead of deriving the Taylor's series by summing infinite derivatives over factorials, I can derive the same approximation from purely geometrical constructs.

I'm wondering if something like this has been done before? If so I don't want to go further and reinvent the wheel. Currently I'm stuck at a certain point because the more accurate you want to get, the complexity of this rises exponentially. But preliminary results are conclusive, this could be true. If you are aware of something like this, let me know. In the mean time I'm going to prepare some papers to show you guys if interested.
 
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  • #2
Just off the top of my head I'm not aware of anything like that, but I'm just a silly undergrad. Could you describe your derivation in more detail?
 
  • #3
You might want to check the "http://www.maa.org/pubs/monthly.html" ] (1970-presesnt) for "visual proofs" or "proof without words" along these lines.
 
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Related to Geometrical interpretation of Taylor series for sine and cosine?

1. What is the geometrical interpretation of a Taylor series for sine and cosine?

The geometrical interpretation of a Taylor series for sine and cosine involves representing these trigonometric functions as infinite sums of polynomials. These polynomials approximate the functions at a given point and converge to the exact value as more terms are added. This can be visualized as a curve approaching the shape of the sine or cosine wave.

2. How is the Taylor series for sine and cosine derived?

The Taylor series for sine and cosine can be derived by expanding these functions around a specific point using their derivatives. This results in a series of terms that can be written in a general form, with each term representing a derivative evaluated at the given point.

3. What is the significance of the radius of convergence in the geometrical interpretation of Taylor series for sine and cosine?

The radius of convergence represents the distance from the center point at which the Taylor series for sine and cosine can accurately approximate the functions. Beyond this radius, the series diverges and is no longer a good approximation. This has implications for the accuracy of the geometrical interpretation.

4. Can the Taylor series for sine and cosine be used to calculate exact values of these functions?

Yes, the Taylor series can be used to calculate exact values of sine and cosine at a given point, as long as the series converges at that point. However, as the number of terms increases, the calculation becomes more complex and computationally intensive.

5. How does the number of terms in the Taylor series affect the accuracy of the geometrical interpretation for sine and cosine?

The more terms that are included in the Taylor series, the closer the approximation will be to the actual value of sine and cosine. However, as the number of terms increases, so does the complexity of the calculation. Therefore, a balance must be struck between accuracy and computational efficiency.

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