### Identities NO. 3 : Trigonometric

30Dec07

For $\alpha + \beta + \gamma = \pi$, we do have
$\tan{\alpha} + \tan{\beta} + \tan{\gamma}= \tan{\alpha} \cdot \tan{\beta} \cdot \tan{\gamma}$
$\tan{\frac{\alpha}{2}} \cdot \tan{\frac{\beta}{2}} + \tan{\frac{\beta}{2}} \cdot \tan{\frac{\gamma}{2}} + \tan{\frac{\gamma}{2}} \cdot \tan{\frac{\alpha}{2}} = 1$\
Trigonometrics

#### 7 Responses to “Identities NO. 3 : Trigonometric”

1. 1 Astrid Denaya Lesa

Really until now, I only know that the definition of tangent function as tan(t)=sin(t)/cos(t). But at last month, I ever visit to http://rohedi.wordpress that present another definition that is in the form :

tan(t) = sin(2t) / [1+cos(2t)

According to the author the above definition was derived from solution of the following ODE

dy/dt = 1 + y^2

with the initial values at t=0 and y=0. The new tangent function appropriates to the common tangent function except for the value t=pi/2 that gives tan(pi/2)=0/0. I ever asked this result to a mathematics forum at address http://castingoutnines.wordpress.com/2007/11/09/i-heart-60s-era-math-books/#comment-17117
but until now they has not yet given the answer. Now, my problem is how to represent tan(t) = sin(2t)/[1+cos(2t) geometrically.

2. 2 masteranza

Geometrically? it looks like a tangent function it’s shaped can be derived from a circle with radius equal to 1.
What I know is that identity works pretty well, since:
$\frac{\sin{2t}}{1+\cos{2t}}=\frac{2 \sin{t}\cos{t} }{\sin^2{t} + \cos^2{t} + \cos^2{t} -\sin^2{t}}= \frac{2\sin{t}\cos{t}}{2 \cos^2{t}}= \frac{\sin{t}}{\cos{t}} = \tan{t}$
and also satisfies the diffrental equotation.

3. After I read how you prove the similarities between the two tangent functions. There are two important things that I can deduce. Firstly, we reminded again of the key success of Chinese people when they succeeded to prove the Pythagoras formula for the first time. What is it? like you, they enter a dummy parameter on a step disprove, here you have used (sint}^2 – (sint}^2. Secondly, we were also rememberred that mathematics “ever” was not exact and it has been saved by L’hospital Theory. Let’s we consider the similarity of

sin(2t)/[1+cos(2t)] =tan(t),

after substituting t by pi/2, he he he…we will find

0/0 = 1/0

Indeed this is very educational posting.

4. The following a part of Planck Formula subject posted on
http://castingoutnines.wordpress.com/2008/12/10/leibniz-on-112/
I think It is also important to be posted here because the new Planck’s formula was created using the solver method of the arctangent differential equation dy/dt = 1+y^2

%%%%%%%%%%%%%
“The discussion about global warming has been commonly associated to the Planck’s formula of Black Body Radiaton. We know that the Planck Formula for the internal energy of black body radiation is of form :

U(T) = ћω / [exp(ћω/(kT) – 1]

But when Einstein verified the phenomenon using harmonic oscillators, the above Planck’s formula can only explain us that the energy difference between the sequence of two energy levels of harmonic oscillator is ћω. But if the internal energy is taken in the following form :

U(T) = 0.5ћω[ tanh{0.5ћω/(kT) – iП/2} – 1]

We will also verify that the minimum energy of the harmonic oscillator is 0.5ћω, and justifying that there are so many harmonic oscillators involved that known from П/2 as the phase differences representing between the two energy lavels. We know that the above of both U(T) is the internal energy of black body radiation in thermal equilibrum. Next question what ’s the form for non thermal equilibrum? Here, I give more information, that there is a possibility to include external perturbation to the differential equation representation of the above New Planck’s Formula.
%%%%%%%%%%%%%%%

Perhaps the New Planck’s Formula that can be useful to explain not only disclose confidential global warming but also in treating the experiment of regulated speed waves. If you and visitors here are interested to know the detail explanation about the alternative method of solving an ordinary differential equation, please visit to this address : http://eqworld.ipmnet.ru/forum/viewtopic.php?f=2&t=34.

5. The following a part of Planck Formula subject posted on
castingoutnines.wordpress.com/2008/12/10/leibniz-on-112/
I think It is also important to be posted here because the new Planck’s formula was created using the solver method of the arctangent differential equation dy/dt = 1+y^2

%%%%%%%%%%%%%
“The discussion about global warming has been commonly associated to the Planck’s formula of Black Body Radiaton. We know that the Planck Formula for the internal energy of black body radiation is of form :

U(T) = ћω / [exp(ћω/(kT) – 1]

But when Einstein verified the phenomenon using harmonic oscillators, the above Planck’s formula can only explain us that the energy difference between the sequence of two energy levels of harmonic oscillator is ћω. But if the internal energy is taken in the following form :

U(T) = 0.5ћω[ tanh{0.5ћω/(kT) – iП/2} – 1]

We will also verify that the minimum energy of the harmonic oscillator is 0.5ћω, and justifying that there are so many harmonic oscillators involved that known from П/2 as the phase differences representing between the two energy lavels. We know that the above of both U(T) is the internal energy of black body radiation in thermal equilibrum. Next question what ’s the form for non thermal equilibrum? Here, I give more information, that there is a possibility to include external perturbation to the differential equation representation of the above New Planck’s Formula.
%%%%%%%%%%%%%%%

Perhaps the New Planck’s Formula that can be useful to explain not only disclose confidential global warming but also in treating the experiment of regulated speed waves. If you and visitors here are interested to know the detail explanation about the alternative method of solving an ordinary differential equation, please visit to this address : eqworld.ipmnet.ru/forum/viewtopic.php?f=2&t=34.

6. 6 andre rizki lesmana

very interesting….

1. 1 Software