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February 28th, 2019, 04:07 AM   #1
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Definition of the angle and pi???

As a University student, I'm re-reading calculus in order to get a full understanding of what is going on and how everything was actually invented (I just love to know how things work).

By reading calculus 1 of Thomas and Finney I found out for the first time that the angle in radians is being defined as followed:

$\displaystyle
\theta = \frac{s}{r}
$

where s is the arc of the circle and r the radius. For a unit circle:

$\displaystyle
\theta = s
$

I can not understand how they came up with this division. I can not visualize how:
$\displaystyle
\theta \cdot r = s
$

will actually give me the length of the arc.
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February 28th, 2019, 05:58 AM   #2
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It's $\theta \cdot r$ that gives the arc length, as the arc length is proportional to both the angle and the radius.
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February 28th, 2019, 06:01 AM   #3
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There are $2\pi$ radians in a circle.
What is the circumference of the circle of radius $r$?
What fraction of this circumference does the arc of angle $\theta$ have?
And therefore, what is the length of the arc of angle $\theta$ and radius $r$?

PS: what you read in current textbooks is not how this stuff was invented - that was generally incredibly messy, with inaccuracies and gaps that were only corrected and filled in later. Textbooks give a sanitised and logical overview of how one might invent this stuff if one knew in advance where one was trying to get to.
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February 28th, 2019, 08:41 AM   #4
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Quote:
Originally Posted by v8archie View Post

PS: what you read in current textbooks is not how this stuff was invented - that was generally incredibly messy, with inaccuracies and gaps that were only corrected and filled in later. Textbooks give a sanitised and logical overview of how one might invent this stuff if one knew in advance where one was trying to get to.
This is actually so true! But this makes me sad because I really want to understand how things were invented.

So in other words, I also need to know and understand how pi was invented in order to understand this formula which connects the angle with the circle's radius and arc?

By the way, can I find any books or other sources that explain mathematics to a more detailed level, including history? How these people exactly planned everything and invented those things. For example, how did they calculate pi? How did they think of calculating it and why in the first place? What is the mathematical proof (with steps)?
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February 28th, 2019, 10:15 AM   #5
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Quote:
Originally Posted by babaliaris View Post
So in other words, I also need to know and understand how pi was invented in order to understand this formula which connects the angle with the circle's radius and arc?
Not really, it's sufficient to know the formula for the circumference of a circle.

Quote:
Originally Posted by babaliaris View Post
For example, how did they calculate pi? How did they think of calculating it and why in the first place? What is the mathematical proof (with steps)?
There are many ways of calculating PI. The first was this: https://www.pbs.org/wgbh/nova/physic...mating-pi.html

It also answers some of your other questions.
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