Question

"Trigonometry is all about triangles (their angles and sides relationships)."


Explain me which (type of) triangle has it's one angle as 270°???
None, right?? Infact, a triangle can have sum of it's all angle as much as 180°.

Then, why things like sin270°, Or Cos180° are in trigonometry???

Cause, we can't have a triangle with an angle 270° or 180°, now how can we get this so called side-angle realtionship of " *a triangle* " if there is no triangle.


*I know how it's derived*, explain me how it makes sense in *practical world* (no theoretical explantions please).

Answer 1

Brief Explaination :

Do the angles of the triangle of a triangle add up to 180° or π radians .
The answer is sometimes yes sometimes no.
Is it a meaningful question?

=> Well no, at least not until we have agreed on the meaning of the words 'angle' and 'triangle'., not until we know the rules of the game .

=> There are axioms that say that sum of the angles of a triangle on the surface of a unit sphere is not equal to π but to π plus the area of the triangle.

Fact : Area of the surface of a unit sphere is 4π .

=>The sides of a triangle ABC are segments of three circles which actually cut the surface of the sphere into 8 spherical triangles.

=>Between two circles through the point A , there are four angles(see pic ).

We label the angle inside triangle ABC as A and similarly other angles as B, and C.

In spherical Geometry, the basic axioms which we assume (the rules of the game ) are different from Euclidean Geometry i.e.Non- Euclidean Geometry.

In spherical geometry, the angles of the triangle do not always sum up to π radians , so we cannot expect the parallel postulates to hold .

Let us take an example :

Suppose, that we want to navigate round the earth, and being a navigator , we want to calculate distances, angles and so on.

=> This means we have to do geometry on the surface of the sphere.
And it is clear that Euclid's Geometry will not work here :

$$$$$$$
Ohhhhh

Now what we can do , that is to change the axioms or invent a new geometry.

Let us invent a new geometry.

=> The path of shortest distance between two points is an arc of a great circle.

In view of this , it now seems reasonable to call the great circles the lines of our new Spherical Geometry.

Consider the triangle on the sphere with vertices (1) at the north pole ,(2) the point where the equatir meets the Greenwich Meridian and (3) the point on the equator with longitude 90°.

=>The arcs of these great circles that join these points should be considered as a spherical triangle for it is made up of three segments of our straight lines in new geometry.

=> However, each angle of triangle is 90°., so now we have a triangle with angle sum of 270° not 180° or π radian.

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1) We have , I suggest reached the point where me must agree that there areat least two different geometries namely Euclidean snd Spherical.

=> They do not contradict each other and niether is right or wrong.
However , thanks to Euclid who sadly never imagined such things .
These Geometries simply represent course if action with differentbstarting points.



Hope you now understand my.answer !

Answer 2

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Hey Mate 

Your questions seem to Amuse me !! Thanks for the Question ^_^

A prior notice : My Answer might not be touching the theories at all !! Its just a matter of ENGLISH and OPINIONS

→ Firstly, 270° is a reflex angle [ Don't know what makes me type that but, maybe it could come in handy while explaining things ]

So, come on along, lets see a few examples :
 → Imagine a Monkey hanging at the Branch, ah ! trust me its relevant !

So, what'd you say, 180° ?? Should it be 180°. a half revolution he should it should be capable of ? I mean, that'd mean he's going ta hit the ground xD

While you CONVERT this whole system as PHYSICS, you're going ta deal with Angular Momentum and stuffs right ? You can't possibly think of such stuffs without the idea of a COMPLETE REVOLUTION being equal to -> 360°
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A 2-D plane deals with Points a lot .... which unfortunately has no existence if you stop unconsciously messing around with an irregular shape of ... probably minute diameter =_=   I mean ... -> ( . ) the thing in the bracket is not actually a POINT  cause ... then, when you say : A Plane has indefinite points, you'll be actually talking of a Black board duh !!

So, what one means while speaking about a point is : 
 " An unconsciously made mark on a piece of paper or on any plane "

What a Point Mathematically is :
 " A non-dimensional imaginary existence, that marks a certain co-ordinate on a plane ... a point of reference used to determine the presence, and location of other things created or present due to it "

( unless you're talking about dihedral angles which ... however change the whole case )

Now, what would you do with a point that only accounts for 50% of its surroundings, i.e., ummm, if you use that for making a triangle, you can't even draw an Exterior Angle or ... an exterior angle bisector... Its like.. you covered a part of that point's surroundings with an infinitely long blanket +_+

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I know, most of what I said... might've gone over head right ??

So lets consider a GRAPH !! The origin, suppose, the Region y < 0 is covered with a sheet of paper ( hypothesized ) !! and hence, the origin has no where to make a 270° Right ? But does that even mean the ratios, cos / sin / tan won't exist ? Cause, there are still indefinite points like in y < 5, y < 10, y < ( + infinity )
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Don't take it the other way !! Not theoretic however, I wish you realize the difference between an angle, the basic existence of it, and the Triangle as well !!!

Angle : is the measurement of the region formed between two rays sharing a common end-point ( the vertex ) 

However when we look at it, Trigonometry cares about Ratios of LINE SEGMENTS, not rays !! Moreover, given the cosine rule, sine rule, etc., one can very subjectively question that : " Trigonometry resides only to a particular limit .. the triangle "

That doesn't change the fact that Its actually the Angles which are utilized as the parameter of these Functions and when we're talking of angles, they're .... they're fractions of a complete revolution around a point !
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Many-a-times, they ask : What is sin ( 720° ) or ... what's sin ( 1440° ) ??
Duh !! Why do they even ask such things =_= well the idea is very subjective !! 

Cause, we deal only with the angles within our reach, only while the Greek Constant THETA falls between the range 0 and 180° ... between 0 and pi radians.. only so that the parallel lines can show the Complimentary and Supplementary properties !! Cause ... wait a sec =_=

I want you toooo read this :
 
" in Euclidean geometry, a point is a primitive notion upon which the geometry is built " 

And henceforth, 

" in Euclidean geometry, an angle is a primitive notion upon which the triangle is built "

Maybe my answer would use the twists and turns to get you to your answer, I want you to understand that its soo relative, the fact that these reflex angles are used, depends upon how the Parallel lines, the Supplimentary property , etc. depends on these angles, and hence, even if they do not have any such practical USE,  its better to get along with it ^_^

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There's this report option underneath !! Hit it if you find it unhelpful xD 
And yeah !! Just as a PCM student has to study Sanskrit for absolutely zero reasoning, Maths student has to rule out reasoning when it comes to Possibilities and Estimations !!

One such question to check out your Hold on Maths might be :

 → "If a + 1/a = 2 Cos 6 then a^1000 + 1/a^1000 + 1 = ??"

Not at all Practical, just to test your HOLD at Understanding the Language of Mathematics ^_^