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#1 2015-12-03 20:13:36

Raj.01
Member
From: Pakistan
Registered: 2015-08-07
Posts: 17
Website

Why unit circle?

Hi all,

Can anyone help me understanding why we always take unit circle while studying trigonometry? What's the glory of unit circle?


Life is the process of narrowing down the probabilities.

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#2 2015-12-04 02:04:50

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,137

Re: Why unit circle?

hi Raj.01

This is what I think:

Trigonometry started with right angled triangles and sine = opp/hyp etc.

But that means you cannot have angles outside the range (0,90), and mathematicians don't like limitations like that so a general definition was needed.  There are two criteria (1) the general definition must be consistent with the right angled triangle definition; and (2) it must extend the idea to any angle, positive or negative.

Now you don't have to have a unit circle for this general definition.  You could choose a circle radius 2 let's say.  You'd have to read off the x and y coordinates of points on the circle making an angle of theta with the positive x axis and then define

and similarly for cos and tan.

But the formulas are simplified if the radius is 1, because then

It's not compulsory but it does make things simpler.

You could make up 'odd' definitions for all sorts of mathematical ideas.  For example, instead of percentages we could have icosages where

The & symbol is the one I've chosen to show it's an icosage.

There's no absolute reason why you shouldn't work always with fractions of 20, rather than 100 as we do with percentages.  Someone could say I got 15& in my exam, where you might say 75%. But we don't because everyone would get very confused by what we were talking about.  And maths has to be something you can communicate to others.  So just accept that the unit circle is the way it's defined and get used to it.  smile

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#3 2015-12-04 07:55:53

Raj.01
Member
From: Pakistan
Registered: 2015-08-07
Posts: 17
Website

Re: Why unit circle?

Thank you for the response.

Actually, this is the question I searched a lot for. I asked so many people, even last week into my class, from my professor. But unfortunately I couldn't get an answer that can satisfy me. So I'm researching on the web, several notes websites etc, and compiling a set of reasons. I must share here once I'm done. Then I'll discuss them with you great people here.


Life is the process of narrowing down the probabilities.

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#4 2015-12-06 02:48:20

Raj.01
Member
From: Pakistan
Registered: 2015-08-07
Posts: 17
Website

Re: Why unit circle?

Hi, can anybody tell me if numbers other than natural numbers existed at the time of Euclid?


Life is the process of narrowing down the probabilities.

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#5 2015-12-06 05:55:28

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,137

Re: Why unit circle?

yes.  Check this out:

https://nrich.maths.org/2515

So fractions yes.  Negatives probably not

https://nrich.maths.org/5747

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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