Pi (π) can be approximated using a limit formula involving trigonometric functions. One such formula is based on the perimeter of a polygon inscribed in a circle.
Calculating Pi with a Limit: x * sin(180/x)
The formula *x sin(180 / x)** can be used to approximate pi. In this formula, 'x' represents the number of sides of a regular polygon inscribed within a circle. As 'x' approaches infinity, the result of the formula approaches pi. To use this formula effectively, ensure your calculator is set to degree mode.
How it Works
Imagine a circle with a regular polygon inside it. As the number of sides (x) of the polygon increases, the polygon more closely resembles the circle. Therefore, the perimeter of the polygon gets closer and closer to the circumference of the circle. The formula xsin(180/x) calculates an approximation of half the circumference of a circle with radius of 1. Since the circumference of a circle is 2πr, with r=1, the circumference is 2π, and half the circumference is π.
Example
Let's say x = 1000:
- Plug in x: 1000 * sin(180 / 1000)
- Calculate: This yields approximately 3.14157.
As you increase x, the result will get closer to pi (approximately 3.14159265359...). This method demonstrates the concept of a limit, where the value of the expression approaches a specific value (in this case, pi) as 'x' tends towards infinity.
Important Considerations:
- Degree Mode: This formula is dependent on the calculator being set to degrees, not radians.
- Approximation: The formula provides an approximation of pi. The accuracy improves as x increases.
- Limits: The formal mathematical representation would involve taking the limit as x approaches infinity: lim (x→∞) x * sin(180 / x) = π
Alternative Limit Formula Using Cosine
Another limit formula to approximate pi is:
2^n * sqrt(2 - sqrt(4 - (P_(n-1)/2^(n-2))^2))where
- P_1 = 2
How it Works
This formula also works based on inscreibing a polygon inside a circle and iteratively calculating its perimeter as the number of sides goes to infinity
Example
- P_1 = 2
- P_2 = 2^(2) sqrt(2 - sqrt(4 - (2/2^(1-2))^2)) = 4 sqrt(2 - sqrt(4 - 4)) = 4 * sqrt(2) = 5.65685424949
- P_3 = ... This calculation quickly becomes tedious but will approximate pi more accurately as n goes to infinity.
In conclusion, the limit calculation x * sin(180/x), with x approaching infinity, provides an increasingly accurate approximation of pi when your calculator is in degree mode. Another, more complex, option exists using cosine instead of sine.
