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February 28th, 2019, 04:07 AM  #1 
Senior Member Joined: Oct 2015 From: Greece Posts: 115 Thanks: 8  Definition of the angle and pi???
As a University student, I'm rereading 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. 
February 28th, 2019, 05:58 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,484 Thanks: 2041 
It's $\theta \cdot r$ that gives the arc length, as the arc length is proportional to both the angle and the radius.

February 28th, 2019, 06:01 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,640 Thanks: 2624 Math Focus: Mainly analysis and algebra 
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. 
February 28th, 2019, 08:41 AM  #4  
Senior Member Joined: Oct 2015 From: Greece Posts: 115 Thanks: 8  Quote:
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)?  
February 28th, 2019, 10:15 AM  #5  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,640 Thanks: 2624 Math Focus: Mainly analysis and algebra  Quote:
Quote:
It also answers some of your other questions.  

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