Question

We know sum of all angel in a triangle us 180 so how sin 270 etc pissible.?

Answer 1

Read my explanation. I expect readers not to edit my explanation. Instead, they could give another, separately.

Explanation:

I take the example of the orbit of the Earth ( E ),

around the Sun ( S ), as reference curve. it is an ellipse.

Its polar equatioh is

`l/r = 1 + e cos theta`, where time-related `theta` is the angle that

SE makes with the recent focal radius SA to A ( about January 21 )

called ( nearest to the Sun ) perihelion...

Here the pole, for [polar coordinates]

(https://socratic.org/trigonometry/the-polar-system/polar-

`( r, theta )` is S at a focus of the orbit,

and the initial line `theta = 0` is SA.

In half a year, E reaches ( farthest from S ) aphelion A'.

AA' is the major axis of the elliptic orbit.

`theta` at A' is `180^o`.

What should be `theta` beyond, for the remaining half year?

There is need for defining

`cos ( 180^o - theta ) = - cos theta`,

so that the equation becomes

`l/r = 1 - e cos theta`

enabling r-reduction, from aphelion SA' to perihelion SA, at the end

of the period of revolution, 1 year.

Then, for the next year, the need is

`cos ( 360^o - theta) = cos theta`, for the first half year, and

`cos ( 540^o - theta) =- cos theta`, for the second, and so on.

In brief,

`cos ( theta + (2k+1 )pi ) = - cos theta` and

`cos ( theta + 2kpi ) = cos theta, k = 0, +-1, +-2, +-3, ...`.

Also, we define here `cos theta` as a periodic function, with

period `2pi = 360^o`.

Not-to-uniform-scale graph:

graph{((x-1)^2/100+y^2/99-1)((x-1)^2/100+y^2/99-0.001)=0[-20 20 -10 10]}]

Here, S is `( 1, 0 ), SA (rarr) = 99.99 and (larr) SA' =10.01`.

The orbit motion is anticlockwise. The dot locates Sun S.