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Euler Definition of Sine

Prerequisites

Cosine is an even function | \(\cos(\theta) = \cos(-\theta)\)
Sine is an odd function | \(\sin(-\theta) = -\sin(\theta)\)
Eulers Formula | \(e^{j \cdot \theta} = \cos(\theta) + j \sin(\theta)\)

Description

This equation defines the sine function (\( \htmlId{tooltip-sine}{\sin} \)) for an angle (\( \htmlId{tooltip-angle}{\theta} \)) in terms of eulers constant (\( \htmlId{tooltip-euler}{e} \)) and the imaginary unit (\( \htmlId{tooltip-imag}{j} \)).

Equation

\[\htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta}) = \frac{\htmlId{tooltip-euler}{e}^{\htmlId{tooltip-imag}{j} \cdot \htmlId{tooltip-angle}{\theta}} - \htmlId{tooltip-euler}{e}^{-\htmlId{tooltip-imag}{j} \cdot \htmlId{tooltip-angle}{\theta}}}{2 \cdot \htmlId{tooltip-imag}{j}}\]

Symbols Used

\(\theta\)

This is a commonly used symbol to represent an angle in mathematics and physics.
\(e\)

This symbol represents Euler's constant. It is approximately \(2.718\).
\(\sin\)

This is the symbol for sine, is a trigonometric function that represents the ratio of the opposite side to the hypotenuse in a right-angled triangle.
\(j\)

This symbol represents the imaginary unit, which is defined as the square root of \(-1\). \( j = \sqrt{-1}\). It is the most fundamental unit in the field of complex numbers, allowing for the expression of numbers that cannot be represented on the real number line.

Derivation

  1. Consider Euler's Formula:
    \[\htmlId{tooltip-euler}{e}^{\htmlId{tooltip-imag}{j} \cdot \htmlId{tooltip-angle}{\theta}} = \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) + \htmlId{tooltip-imag}{j} \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta})\]
  2. Let us also consider a version of Euler's Formula where we use \(-\htmlId{tooltip-angle}{\theta}\) instead:
    \[\htmlId{tooltip-euler}{e}^{\htmlId{tooltip-imag}{j} \cdot - \htmlId{tooltip-angle}{\theta}} = \htmlId{tooltip-cosine}{\cos}(-\htmlId{tooltip-angle}{\theta}) + \htmlId{tooltip-imag}{j} \cdot \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta})\]
  3. We can now simplify using the fact that sine is an odd function and cosine is an even function:
    \[\htmlId{tooltip-sine}{\sin}(-\htmlId{tooltip-angle}{\theta}) = -\htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta})\]
    \[\htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) = \htmlId{tooltip-cosine}{\cos}(-\htmlId{tooltip-angle}{\theta})\]
    to get...
    \[\htmlId{tooltip-euler}{e}^{-\htmlId{tooltip-imag}{j} \cdot \htmlId{tooltip-angle}{\theta}} = \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-angle}{\theta}) - \htmlId{tooltip-imag}{j} \cdot \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta})\]
  4. We can now subtract this from Euler's Formula (see step (1)) to get
    \[\htmlId{tooltip-euler}{e}^{\htmlId{tooltip-imag}{j} \cdot \htmlId{tooltip-angle}{\theta}} - \htmlId{tooltip-euler}{e}^{-\htmlId{tooltip-imag}{j} \cdot \htmlId{tooltip-angle}{\theta}} = 2 \cdot \htmlId{tooltip-imag}{j} \cdot \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta})\]
  5. Finally, we can divide both sides by \((2 \cdot \htmlId{tooltip-imag}{j})\) to get...
    \[\htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-angle}{\theta}) = \frac{\htmlId{tooltip-euler}{e}^{\htmlId{tooltip-imag}{j} \cdot \htmlId{tooltip-angle}{\theta}} - \htmlId{tooltip-euler}{e}^{-\htmlId{tooltip-imag}{j} \cdot \htmlId{tooltip-angle}{\theta}}}{2 \cdot \htmlId{tooltip-imag}{j}}\]
    as required...

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