What is the value of pi in trigonometry?

A Brief History of π Pi has been known for almost 4000 years—but even if we calculated the number of seconds in those 4000 years and calculated pi to that number of places, we would still only be approximating its actual value.

Pi is defined as the ratio of the circumference of any circle to its diameter. The perimeter (circumference) of an infinitely sided polygon (circle) divided by the diameter is ALWAYS the same number. A mathematical constant is a number that does not change.

It is a number that cannot be written as a ratio of two integers (or cannot be expressed as a fraction). For example, the square root of 2 is an irrational number because it cannot be written as a ratio of two integers. Pi is an unending, never repeating decimal, or an irrational number.

So let’s take a closer look at the sine and cosines graphs, keeping in mind that tan(θ) = sin(θ)/cos(θ). The tangent will be zero wherever its numerator (the sine) is zero. This happens at 0, π, 2π, 3π, etc, and at –π, –2π, –3π, etc.

Example: π (Pi) is a famous irrational number. We cannot write down a simple fraction that equals Pi. The popular approximation of 22/7 = 3.1428571428571 is close but not accurate. Another clue is that the decimal goes on forever without repeating.

Cos(2 pi) is equal to 1. The cosine function, cos(x), oscillates between 1 and -1 with a period of 2pi as x varies. By definition, cos(0) = 1, and the periodicity of the function means the cosine of all multiples of 2pi (2pi, 4pi and so on) is also equal to 1.

Hence, neither does pi/2. A degree is defined such that exactly 360 form a circle. A radian is defined as the angle at which the length along the edge of the circle is equal to the radius of the circle. pi is the number of radians in one-half of the circle.

The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse).

When starting off in math, students are introduced to pi as a value of 3.14 or 3.14159. Though it is an irrational number, some use rational expressions to estimate pi, like 22/7 of 333/106. These rational expressions are only accurate to a couple of decimal places, however.

Because π is irrational, it has an infinite number of digits in its decimal representation, and it does not settle into an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique.

The record for finding consecutive numbers, from 3.14 onward to the final digit, is held by Fabrice Bellard, who announced in January that he had calculated pi to 2.7 trillion digits.

Proof that 22/7 exceeds π Proofs of the famous mathematical result that the rational number 227 is greater than π (pi) date back to antiquity. If one knows that π is approximately 3.14159, then it trivially follows that π < 227, which is approximately 3.142857.

Because while these other national holidays come to an end, Pi Day actually doesn’t come to an end, because though Pi technically isn’t infinite, it does, in a sense, never fully end. Pi, formally known as π in the world of mathematics, is the ratio of the circumference of a circle and the diameter of a circle.

The sequence of natural numbers never ends, and is infinite. There’s no reason why the 3s should ever stop: they repeat infinitely. So, when we see a number like “0.999” (i.e. a decimal number with an infinite series of 9s), there is no end to the number of 9s.

Pi is an irrational number, which means it cannot be represented as a simple fraction, and those numbers cannot be represented as terminating or repeating decimals. Therefore, the digits of pi go on forever in a seemingly random sequence.

Pi is an irrational number. As such, it has no final digit. Furthermore, there is no pattern to its digits. But for what it’s worth, here are the first thousand digits of pi.

In 1706 a little-known mathematics teacher named William Jones first used a symbol to represent the platonic concept of pi, an ideal that in numerical terms can be approached, but never reached.