### Theory NO. 3 : Geometry, tangents and angles in circle

26Jun08

The angle which arms contain chords or tangents of a circle is the half of the sum (if the verticle of the angle lies inside the circle or on it) or difference (if the verticle lies outside) of middle angles based on the curves, which set this angle.

The proof can be illustrated by following pictures which consider all the possibilities.

#### 4 Responses to “Theory NO. 3 : Geometry, tangents and angles in circle”

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 sagar

sin2t= 2sint cost; 1+cos2t= 2cost cost

2. 3 Denaya Lesa

Hi Masteransa, are you forget me?

Maybe you are interested to discuss my post on the website.

3. Hi Masteranza,

Oh yeah, if you would look the nice number besides exp(i*pi)+1=0 from Leonhard Euler, please click Denaya Lesa’s address. At the link it has been posted the nice “Pi(Phi)” number, that presents an expression of the Pi exact formula as function of Phi Golden Ratio.

Bye…bye…
Denaya Lesa.