Thursday, October 27, 2011

Trigonometry

When we talks about trigonometry, we will most probably think of trigonometric functions, e.g. sine, cosine, tangent etc. Ever think of how were the graphs of these functions obtained?

Let's consider a right triangle with hypotenus of 1 unit. You may drag the point C below and observe the change of the angle. Recall that

$\sin\theta=\frac{y}{r}$
$\cos\theta=\frac{x}{r}$
$\tan\theta=\frac{y}{x}$

Point E is (x, sin x), point F is (x, cos x) and point G is (x, tan x). Observe the position of point E, F or G as you drag point C. You may choose the desire graph at the right bottom corner of the graph.

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If you wish to know more about trigonometric functions, please click on the function you are interested under Trigonometry at the left column.

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Friday, October 21, 2011

Hyperbolic Tangent & Hyperbolic Cotangent

Hyperbolic tangent is defined as

$\fn_phv \tanh(x)=\frac{\sinh(x)}{\cosh(x)}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$,

while hyperbolic cotangent is the reciprocal of hyperbolic tangent, and it is defined as

$\fn_phv \coth(x)=\frac{\cosh(x)}{\sinh(x)}=\frac{e^x+e^{-x}}{e^x-e^{-x}}$.

If we want to sketch the graph of the earlier function, think of the y-intercept and limits below:

If we want to sketch the graph of the later function, think of the limits below:

The concept above applied for all values of a > 0. You may slide the value of 'a' below to look at the changes of $\fn_phv \tanh(ax)$ and $\fn_phv \coth(ax)$ for different values of 'a'.

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Betty, Created with GeoGebra

Hyperbolic Cosine & Hyperbolic Secant

The hyperbolic cosine function is defined as
$\cosh(x)=\frac{e^{x} +e^{-x}}{2}$

or generally,
$\cosh(ax)=\frac{e^{ax} +e^{-ax}}{2}$.

Look at the graph below, try to slide the value of 'a' to different values to view the changes on graph.

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Now, take a look at the graph of the reciprocal of hyperbolic cosine function, i.e. hyperbolic secant function, which is defined as
$\textup{sech}(ax)=\frac{1}{\cosh(x)}=\frac{2}{e^{ax} +e^{-ax}}$

Observe that when a = 1,

Again, slide the value of 'a' to view the changes of the graph.

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Hyperbolic Sine & Hyperbolic Cosecant

The hyperbolic sine function is defined as
$\sinh (x)=\frac{e^x -e^{-x}}{2}$

or generally,
$\sinh (ax)=\frac{e^{ax} -e^{-ax}}{2}$

Look at the graph below, try to slide the value of 'a' to different values to view the changes on graph.

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Now, take a look at the graph of the reciprocal of hyperbolic sine function, i.e. hyperbolic cosecant function, which is defined as
$\textup{csch} \; (ax)=\frac{1}{\sinh{ax}}=\frac{2}{e^{ax} -e^{-ax}}$.

Observe that when a =1,
Again, slide the value of 'a' to view the changes of the graph.

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