# 4 * sin(ϕ * 32)(A new theorem in development)

• greggory
In summary, the conversation revolved around a theorem about properties of a circle. The person shared their work on the theorem, including a link to an image. They noticed that the value of sin(45) was close to π/4 and continued to increase the value until they reached sin(51), which was also close. They then tried using a method from Archimedes to find a more accurate value for π, but this did not work. They then tried adding decimals after 51 and found that sin(51.7575) * 4 was equal to 3.14159. They also noticed a connection with the value ϕ * 32, which gave them a value of 3.142438. They speculated
greggory
So, I was working with a theorem that I had been working on about more properties of a circle, more invisible properties.

Here is my work for the theorem. It isn't much, but it is something:

http://img6.imageshack.us/img6/8938/theorem.png

I noticed that when I did sin of 45, I got the value 0.707106, which I knew was close to π / 4, so I continued to increase the value. When I got to sin of 51, I got 0.777, which was really close, so I tried 52. Unfortunately, it would have been 3.15, which I knew wasn't π, but I knew something. I tried what(if I remember who it was) Archemidies, who specified a values that was higher and lower than pi. So, I did sin(51) < π < sin(52). I then began adding decimals after 51. I got to sin(51.7575) * 4, which was equal to 3.14159. I then noticed that ϕ * 32 was similar to this value. So, I replaced the value with ϕ * 32, which is approximately 51.777, and I got the value 3.142438 approximately.

From here, I noticed that maybe this formula could have calculate pi faster than any other method before, because when I tried using the Fibonacci numbers to calculate phi for this formula, one time the value was close to 3.14159265, ect, but it kept going, then when it reached 3.1424 it stopped and went from there.

Does this mean anything?

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greggory said:
So, I was working with a theorem that I had been working on about more properties of a circle, more invisible properties.

Here is my work for the theorem. It isn't much, but it is something:

http://img6.imageshack.us/img6/8938/theorem.png

I noticed that when I did sin of 45, I got the value 0.707106, which I knew was close to π / 4, so I continued to increase the value. When I got to sin of 51, I got 0.777, which was really close, so I tried 52. Unfortunately, it would have been 3.15, which I knew wasn't π, but I knew something. I tried what(if I remember who it was) Archemidies, who specified a values that was higher and lower than pi. So, I did sin(51) < π < sin(52).
You can stop right here, as this is manifestly not true. For any real number x, -1 <= sin(x) <= 1.
greggory said:
I then began adding decimals after 51. I got to sin(51.7575) * 4, which was equal to 3.14159. I then noticed that ϕ * 32 was similar to this value. So, I replaced the value with ϕ * 32, which is approximately 51.777, and I got the value 3.142438 approximately.

From here, I noticed that maybe this formula could have calculate pi faster than any other method before, because when I tried using the Fibonacci numbers to calculate phi for this formula, one time the value was close to 3.14159265, ect, but it kept going, then when it reached 3.1424 it stopped and went from there.

Does this mean anything?

What you seem to be doing is attempting to solve the equation 4 sin(x) = $\pi/4$. A solution to this equation is x = sin-1($\pi/4$) ≈ .903339 radians ≈ 51.75°.

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It is true that, for x measured in radians, $\lim_{x\to 0}sin(x)/x= 1$. That is, for x very close to 0, sin(x) will be very close to x. Once you get up to things like 45 degrees, which is equivalent to $\pi/4$ radians, they are still "close" but not "very close". The farther from 0 you get, the farther apart those get.

@Mark44
OH, thank you for the contribution. I never saw that there.

Edited.

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I am intrigued by your discovery and the potential implications it may have. However, it is important to note that this is still a work in progress and requires further testing and analysis before it can be considered a new theorem. It is also important to consider other factors that may have contributed to your findings, such as rounding errors or coincidences. I would suggest conducting further experiments and research to validate your results and determine the significance of this formula. This could potentially lead to a new understanding of the properties of circles and could have practical applications in various fields of science and mathematics. Keep exploring and documenting your findings, and don't hesitate to collaborate with other scientists to further develop this theorem.

## 1. What is the purpose of developing this new theorem?

The purpose of developing this new theorem is to expand our understanding of mathematical principles and potentially discover new applications of the sine function in different contexts.

## 2. How is this theorem different from existing theorems involving sine?

This theorem introduces a new factor of 32 in the argument of the sine function, which may result in different patterns and behaviors of the function compared to existing theorems.

## 3. What is the significance of multiplying by 4 in this theorem?

The factor of 4 may have a significant impact on the amplitude and frequency of the sine function, leading to interesting and potentially useful results.

## 4. Is there any practical application for this theorem?

At this stage, the practical applications of this theorem are still being explored and developed. However, it may have potential uses in fields such as engineering, physics, and economics.

## 5. How can this theorem be proven or tested?

This theorem can be proven by using mathematical techniques such as induction, proof by contradiction, or direct proof. It can also be tested by plugging in different values for the angle ϕ and analyzing the resulting patterns and behaviors of the sine function.

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