I already know this, but Eddie is so good at teaching that I’m watching his videos to find good nuggets for my kid. I had the exact same reaction at that moment. Eddie. You are a phenomenon.
It seems to me, that the amazement in the end was based implicitly in the realization, that for any circle the ratio of circumference to diameter is a constant, namely pie. Then again I've not been explicitly aware, that one radian is one radius of length on the arc. Feels like two insights for one mathematical truth :)
This person, in what seems like 3 videos totalling around 30 minutes, taught me that thing, which over 3 YEARS and 7 different teachers later I still couldn't understand. Why? All of them basically just repeated the definition and were done with the explaining. This man went into much more detail than I could have expected. Thank you.
I got what radian was as a definition but it was extremely hard for me to visualize it and see it like degrees but after watching this video I have been enlightened. I can't thank you enough sir.
Good little series ... it is now apparent why the series definitions of sine and cosine would naturally be in radians - it is all talking about ratios !
This is first RU-vid math teacher I've seen explain what the essence of a radian really is: a radian angle measurement is merely the RATIO between the length of the radius of the circle and the length of the arc traced out by that angle on the circle. If the two lengths are equal then we have a 1:1 ratio and the angle between the starting point of the arc and it's end point is 1 radian. If the ratio between the arc length and the radius length is 3.14:1 then the angle is 3.14 radians or pi radians or 180 degrees. We're using the radius length as our UNIT OF MEASUREMENT. If the radius is 7 inches or 627 miles we're mentally making it a 1 on our unit of measurement for simplicity's sake. At the end we can multiply our radian result times the 7 inches of radius or the 627 miles of radius to get our real world measurements of the length of the arc in tape measures or on our odometers respectively. The milky way is 100,000 light years across and therefore has a radius of 50,000 light years. If we drive a starship a quarter ways around the rim of our galaxy the angle between the two points will be pi/2 or 1.57 radians. The arc traversed will be 1.57x50,000 light years in length.
I only just found you recently but all this high level trigonometric learning that my class hasn’t done really helps! Thanks so much - I really enjoy maths and understanding it, which you are great at stimulating.
I think an explanation of the origin of pi would help a lot with clarity here. As in, the distance around the circumference of the unit circle, anti-clockwise from 0π to π ( 0˚ to 180˚ degrees), is 3.14 radians, which is of course, the numerical value of pi.
pi are circumference by diameter which is always constant. They just made a symbol for it, or we could have it as k and write C=2kr and of course since k is constant found experimently by drawing circles and taking ratios k=3.14 just replace k with pi and there you go.
@@ac17dollars It is very difficult to get pi measured accurately, when do it by experiment. An ancient idea for how to calculate pi is to develop an equation of a regular n-gon (as in a polygon with n equal sides and n equal angles), and calculate the ratio of its perimeter to its diameter. Take n as high as practical to calculate, and you can determine how to calculate pi. The n-gon has an area that can be derived from breaking it into triangles. This is what Archimedes did, with the 96-gon. Newton came up with a much more computationally efficient method for calculating pi, based on integrating sqrt(1-x^2) as an infinite series.
By this logic, if you coined a hypothetical angle unit called the diadian (1*pi diadians = 1 revolution), that was defined such that it would be the arc length of 1 diameter wrapped around the circle per unit diameter, it would also "not be a unit" just like the radian. Yet if both these concepts coexisted, we'd still have to call at least one of them a unit.
pi radian= 180 degrees. now taking 180 on the other side, we have pi/180 radians = 1 degree. now radian is a unitless quantity, so any number attached to it must be a unitless number. thus pi/180 radian is a unitless number. so following the above equation , 1 degree is a unitless number. now if degree were not a unitless quantity, it would mean that 1 degree is not a unitless number. (contradiction) thus, degree is a unitless quantity as well. is my argument accurate?
No. The problem with it is saying that pi = 180 degrees. In truth these aren't equal in the mathematical sense. They are two different ways to express the same idea, but that doesn't make them mathematically equal. Therefore you can't really do things like divide both sides by 180. Writing it that way is shorthand to allow new students to convert from the degree system to radians, but it's inaccurate. The most you can say is that 180 degrees corresponds to pi radians, but they aren't equal.
I'm always amazed at the rudeness of the Western students who are talking and making such noise in the backgrounds - the etiquettes with the teacher are very poor, interrupting him, talking to each other while he's talking... He even is so patient with them, but they don't appreciate it.
"a dimensionless quantity" student in background "WHAT!?" me "lolololololololol" in case you are in the same place a radian is the radius of the given circle which can increase or decrease in size, using the unit circle in fact makes this a constant when you really think about it.
The length of the arc between the starting point and ending point of the angle divided by the length of the radius. If the length of the arc is 56.4 centimeters and the radius of the circle is 56.4 centimeters then WE SAY that the ANGLE (not the length) between the starting point and the end point of that arc is 1 radian. If the arc length is 6452 miles and the radius length of this great big circle is also 6452 miles then the ANGLE between the two points of the arc is also 1 radian. A radian is thus a RATIO and not so much a real world measurement. We multiply our resulting radians times the real world length of the radius to get the real world length of our arc.
Because everyone else who forgot half their high school math, still thinks in degrees for angles. Plus, most tools for angles are calibrated in degrees. If you want to communicate with people who will build what you design, you specify your angles in degrees. Tell a machinist or a carpenter to build a pi/4 angle between two lines in your design, and that is Greek to them in more ways than one. You can find radian protractors, but good luck finding a radian miter gauge. Radians exist to make Calculus and Physics as elegant as they can be, degrees exist to communicate with everyone who isn't familiar with radians.
