Word of the Day

## Thursday, November 8, 2012

### Just Decided We Need MathJax

Now we can use LaTeX notations to display mathematic symbols and expressions in our Linguae Textae.
For inline maths, use the dollar sign to open AND close them.
For display maths (an independent line), use '\' immediately followed by '[' to open a math and use '\' immediately followed by ']' to close it.

Inverse proportionality is a relation between two numbers $x$ and $y$ ($x, y\in \mathbb R$) such that:

$xy = k$

Using a parameter $\theta$, any point on the relation $xy = k$ can be represented by: $(\cos\theta, \sin\theta)$

This gives the following parametric equation for inverse proportionality:
$\cos\theta\cdot\sin\theta = k$

If the shape of inverse proportionality on the Cartesian plain is tilted for $-45^\circ$ (clockwise rotation), any point on the resultant shape satisfies the following:

$\cos(\theta+45^\circ)\cdot\sin(\theta+45^\circ) = k$

The left of the equation is:

$= (\cos\theta\cdot\cos45^\circ-\sin\theta\cdot\sin45^\circ) (\sin\theta\cdot\cos45^\circ+\cos\theta\sin45^\circ)$
$= {\sqrt2 \over 2}(\cos\theta-\sin\theta) (\cos\theta+\sin\theta) (\because \cos45^\circ = sin45^\circ = {\sqrt2 \over 2})$

Thus the whole equation is:

$\cos^2\theta - \sin^2\theta = \sqrt2k$

This is identical to the graph of unit hyperbola magnified to a constant ($\sqrt2k$). We know from this that inverse proportionality can be mapped to hyperbola by simple rotation!