Math is like a beautiful, truthful story. However, you need a real story teller, that has passion and loves the story he's telling you to at least grasp the moral of the story, to make you say: tell me another one! Tell it again! Eddie Woo....you do that. Thank you.
radians are great they are handy in physics. a car with a wheel of 1 foot and rotating at 100 radians per second the car will be moving at 100 feet per second V=Rc R is radius c is radions per second V= (radius units) metres or ft per second
To know the velocity of a body in orbit you use v=wr, where r is the radius and w is the angular velocity in rad/s. The first time I heard about them they seemed useless, but theyre actually great
I always learned what radian is, and months later I forget about it. Then I watch again, forget, watch, forget... But thanks for your video to help me catch up in just a couple minutes :)
As a mathematics tutor your videos are very nice and give that extra insight which makes you a very good teacher. Please make a playlist of all these cool videos of understanding as I really enjoy the simplicity in the understanding especially when its linked the basic concepts. Keep up the great work!
quite clear. But could you show me some mathematical history about the introduction of radians, such as what problems drove the mathematician to look for another way of angle measurement, and what kinds things directed the mathematician to the idea of measuring the arc length? Thanks a lot.
One thing to consider -- the ancient Chinese in astronomy used 360 degrees, which spread Westward. This system was adopted for circumnavigating the oceans, first by Zheng He, then by the European explorers. In ancient Greek math, there were only INTEGERS and rational numbers for measuring -- the square-root of 2 caused a major meltdown. That said, Babylonians, Egyptians, and Greeks knew that a rope cut to the length of a circle's radius could measure off the circumference at somewhat more than 6 lengths (or 3 for a semicircle). In this way, the ancients laid the conceptual foundation for radians with a length of rope -- but lacked the numerical sophistication to take it further. Degrees were so deeply successful in navigating the globe, they went unchallenged until the early 1700s, when Roger Cotes laid the modern notion of radians and their use in trigonometry.
the area of the segment formula intuitively leads to the conclusion that theta ~= sin(theta) for small angles (expressed in radians) where the segment area seems to approach limit zero PS: appreciation of this fundamental topic grows manifold after studying applied rotational dynamics
This is also an issue in computer graphics APIs. It is true that humans find angles in degrees easier to interpret and specify, even though the underlying calculations are usually easier to perform in radians. I think the simplest approach is to specify all angles in radians, and convert from degrees for user input and back to degrees for user output. Sometimes it can be convenient for the user to work in other angle units as well, for example units of a whole circle, where 1 circle = 360° = 2π radians. It is common for maths libraries to provide functions to perform the necessary conversions. But then you need two functions for each angle unit, one to convert to radians and one to convert from radians. It is simpler to provide a single conversion factor for each unit, for example for degrees you could provide the constant “deg” defined as follows: deg = 2 * math.pi / 180 Then if a value “theta” is in degrees, it can be converted easily to radians, for example to compute the sine, thus: s = math.sin(theta * deg) and conversely a value in radians can be easily converted to degrees by dividing by the same factor: theta = math.asin(s) / deg
In robotics for revolute objects, I use "revolutions" ("revs") as the preferred units. It's very intuitive. A quarter turn is 0.25 revs. The fact that it's not taught in high-school math might indicate how little applied math experience the teacher has.
@@pbierre Yup, that would be easy to do. In fact, I already have a unit called “circle” which is the same as yours. In my scheme, each new unit only needs you to define an appropriate conversion factor. Simpler than having to define separate “to-radians” and “from-radians” functions, don’t you think?
arc length at 1 radians = r arc length at θ radians (θ = 180°) = θ * r arc length = s = θ * r also arc length of semi circle is = π * r, hence >> π * r = 180° * r >> π = 180°
When I was taking HS mathematics radians was never explained to me like this. We were given the unit circle and told to memorize it. It never made sense at the time.
I guess that's what got me uninterested in maths because for a lot of my lessons (and most of my school life), I was just told to "memorize the formula/equation". It was never explained WHY these formulas/equations even exist or are written the way they are. I was never taught to understand maths, but to just take it as it is. So it felt like something that was devoid if any life, which is funny considering it's everywhere in our lives. So with that in mind, I wanna thank you immensely, Mr. Woo for uploading these videos and reigniting my interest in mathematics. Because you give it life.
Radians exist for one reason - the make the slope of the function y=sin(x) equal to 1 at x=0. This makes the derivative of the sine the cosine and, in turn, the derivative of cosine equal to negative sine. The rest of the trig derivatives follow. When the period of y=sin(x) is 360, the slope at x=0 is pi/180 which is not very convenient.
That's just the tip of the iceberg. When people first wanted to come up with ways to compute sine functions (e.g. Taylor series summations), there was an obvious advantage to computing sin(x) with x in radians.
01-07-2022 Bangalore Sir, What is the purpose of knowing radian measurement? One complete circle is equal to 360 degrees. Whereas, 6.28 radians are equal to 360 degrees, one complete circle. What is the use of radians? Clarify. - Nagaraj T V
I can imagine a flex-tape the exact length of the radius, and marked like a ruler into 10ths, 100ths and 1000ths. When I use this to measure the circumference arc of an angle, the readout number will give the angle width in radians.
Because the calculus of trig functions is as elegant as can be, when you use radians. The cycle of derivatives of sine, as you proceed to higher orders of differentiation follows a 4-part cycle. d/dx sin(x) = cos(x), d/dx cos(x) = -sin(x), d/dx (-sin(x)) = -cos(x), d/dx (-cos(x)) = sin(x) back where we started. If you use degrees as an angle unit, you'll accumulate a pi/180 factor every time you take a derivative, which complicates the matter.
He made the idea of arc length too confusing. Example of not meeting the student where they are. He should have simply said this: The arc length is the radius times the number of radius' you have traveled. Next we know 1 radius is equivalent to 1 radian proportion wise. So we then can simply replace the number of radius' by the number of radians. So we get Length = r X radians
This guy should be teaching this level math. And the one ahead. And the one ahead. Actually i want him to write a book for all level of mathematics, bc then i can just understand my 400 level analysis course.
