Author(s): Sami Almuaigel
In this study, we present a novel method for determining the mathematical value of pi by utilizing the relationship between pi and the tangent function in trigonometry. By examining the relationship between the circumference of a circle and a right-angled triangle, we can derive a new equation for pi, that can be used to simplify the mathematical and physical equations that involve pi.
Pi, denoted by the Greek letter ?, is a mathematical constant widely used in mathematics and physics. It is defined as the ratio of the circumference of a circle to its diameter and, has a value of approximately 3.14159. The concept of pi has been known since ancient times, and has been studied by many mathematicians. Despite its importance, pi is an irrational number, meaning that it cannot be expressed as the ratio of the two integers. The note angle was measured in degrees.
According to the equation for the circumference of a circle, 2?r, if the circumference is visualized as a straight line, the length of the straight line would be 2?r. If we now imagine that the radius of circle r is perpendicular to this straight line, we would have the shape of a right-angled triangle.
A question arises, is there a relationship between this triangle
and the circle?
We assume that the area of this triangle is equal to the area of
It is apparent that if the radius of the circle is fixed and the circumference of the circle represented by a straight line in the triangle is divided into half, quarter, or smaller segments, the equation takes the following form:
We will get Multiple angle values were obtained by varying n. We can now incorporate ? into laws and equations containing trigonometric functions and similarly include trigonometric functions in laws and equations that involve ?.
We established a relationship between pi and the tan angle. However, we aimed to simplify the law connecting them. To achieve this, we first impose the relationship in the following manner.
We know from Euler's identity equation
We can convert any non-zero real number, whether positive or negative, into a number related to pi and the tangent.
Example 8: number 5 can convert to:
Derived in this study can also be used to simplify the mathematical and physical equations that involve pi by replacing pi with the equivalent value derived in this study and converting any number into a number related to pi and tangent.
The study did not receive any financial support from the Foundation of Basic Research or any research program. Author Sami Almuaigel did not receive any funding from the Ministry.
All procedures involving human participants in this study were conducted in accordance with the ethical standards of the institutional and/or national research committee, as well as the 1964 Helsinki Declaration and its later amendments, or comparable ethical standards.
