Tau (τ), Pi (π) and Eta (η)
Tau = pi/2 = eta/4
There are 3 fundamental constants to work out various characteristics of a circle. The three main ones are:
Tau (τ): 2 π. Tau is useful for working out circular trigonometry and also working out numbers on complex planes. Tau is argued to be the actual replacement for pi because the circumference of a circle must be measured using the radius a it is the radius that is the fundamental part of the circle that gives the circle it’s properly.
Pi (π): 3.142592653589793238462643383… The number is irrational and represents the ratio of the circumference of a circle when compared to the diameter. Pi is more useful than Tau when working out the area of a circle. The area of a circle: A=πr^2
Eta (η): π/2. Eta is more useful when working the right angled trigonometry when using a circle. Eta is also significant in working out the area of a circle using a square within it or vice versa.
This article is about Eta and the properties of Eta.
Eta= Area of a circle ÷ Area of a square (as shown above). In the example above, the line AB is the diameter of the circle and is also the longest side (Hypotenuse) of the square.
However, that is not the only interesting thing about Eta.
Eta is equal to a quarter of a circle, so Eta is equal to 90°.
The unit circle helps us use the trigonometric ratios of Sin and Cos. We can use Eta to relate to those trigonometric ratios in such a way that we can write Sin in terms of Cos and vice versa.
Looking at the triangle above, we know that
Sin (θ = a/c (Opposite divided by hypotenuse)
Cos (θ = b/c (Adjacent divided by hypotenuse)
This means that we can arrange the ratios to make Sin (η-θ) = b/c or Cos (θ
It also means that Cos (η-θ) = a/c or Sin (θ.
Sin (η+θ) = Sin (η-θ) = Cos (θ
Sin (2η+θ) = -Sin (θ
Sin (3η+θ) = - Cos (θ
Sin (4η+θ) = Sin (θ
We can also relate η to Euler’s formula.
e^2iη= e^iπ = -1
e^4iη= e^2iπ= e^iτ= 1
In this case, the even exponentials are real numbers, however; the odd ones are complex/imaginary numbers.
Moreover, there is another good theory we can relate Eta into: The Needle theory:
Here, the probability of a needle landing on a parallel line if thrown on the grid is 2/ π or 1/η.
These explanations show that the use of Eta instead of Pi or Tau to make right angled trigonometry easier to understand. These papers also argue for the fact that Eta should be the Fundamental Circle constant as Eta can be more widely used than Tau or Pi however in some cases, Tau and Pi are easier to use. Overall, I’m concluding that students use their common sense to work out which constant is the best to use and makes more sense when working with a certain constant.
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