@@murrrkkkif you want more of that vibe, try listening to the nurse character from nyan neko sugar girls, but beware, the animation is NOT nice and is probably made on ms paint
An easier way to demonstrate why the rectangles and triangles are related is to slide them over to make parallelograms of the rectangles first / make the triangle first then slide the tip Echoing what people have said about audio quality. As for the rest of the video I really enjoyed the presentation and I really look forward to see what’ll come of it! Always happy to come across quality maths channels :)
Haven't watched the video, here's my solution: Let's take radius of the circle as a unit. Let's say the arc starts at angle a1 and ends at angle a2, a1
Nice problem; and very elegant solution. I approached it algebraically. With θ as you've assigned, let σ and τ be the 1st quadrant arcs above and below s respectively. Then A = ½ ( σ - sin(σ)cos(σ) ) - ½ ( τ - sin(τ)cos(τ) ) = ½ [ (σ - τ) - ( sin(τ)cos(τ) - sin(σ)cos(σ) ) ] = ½ θ - ½ ( sin(τ)cos(τ) - sin(σ)cos(σ) ). Likewise B = ½ θ - ½ ( sin(σ)cos(σ) - sin(τ)cos(τ) ). Summing then gives A + B = θ = length(s) as required. Not as elegant - but I believe also reachable by a young audience. I'll keep an eye out for your content - but won't subscribe until you get the audio fixed up.
Good stuff. I call this base 1 math. Because of exponentiation, it wouldn't reduce to Theta if the radius were any larger. (But it would be 2x) But this same principle is how calculus and a circle's area works. The curve determines its area. It's also why Heron's Formula doesn't work that well outside of Base 1. But a+b+c=a*b*c is always a triangle. No matter how you calculate it, but the most elegant nuances are found in base 1.
One thing is still disturbing me, that is, wether this equation is dimensionally equivalent. Because one side is area and the other side is (arc) length.
Honestly great video, I remember struggling on this problem before giving up after a few hours. About the mic quality. I think your setup is likely fine, just needs two things. One, move away from the mic by probably ~6 inches. Second, dont direct your mouth directly at the mic (this is fine with some, but probably not with yours), the main 'gust of air' coming from your mouth should pass just above the mic. This will prevent the sort of super intense sound you get with "p" or "k" sounds. Good luck and cheers!
A more straight forward approach would be to simply find each of the areas using integration and adding them. take coordinates of the points as (rcosalpha,rsinalpha) and (rcosbeta,rsinbeta) and the curve is x^2 +y^2 =r^2. Integrating and adding gives r^2 (beta-alpha) =r*arc length
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Gotta increase sound quality. Animations are nice though.
The sound gives off a crazy vibe tho
@@murrrkkkif you want more of that vibe, try listening to the nurse character from nyan neko sugar girls, but beware, the animation is NOT nice and is probably made on ms paint
An easier way to demonstrate why the rectangles and triangles are related is to slide them over to make parallelograms of the rectangles first / make the triangle first then slide the tip
Echoing what people have said about audio quality. As for the rest of the video I really enjoyed the presentation and I really look forward to see what’ll come of it! Always happy to come across quality maths channels :)
this was glorious the whole way through
I would just probably integrate
Bruteforce way
Haven't watched the video, here's my solution:
Let's take radius of the circle as a unit.
Let's say the arc starts at angle a1 and ends at angle a2, a1
That was good ... too good explaination 😌
Your mic is amazing 🤩
amazing video! thanks a lot! keep up the good work!
great video!
Amazing video, thank you!
Nice problem; and very elegant solution.
I approached it algebraically. With θ as you've assigned, let σ and τ be the 1st quadrant arcs above and below s respectively. Then
A = ½ ( σ - sin(σ)cos(σ) ) - ½ ( τ - sin(τ)cos(τ) )
= ½ [ (σ - τ) - ( sin(τ)cos(τ) - sin(σ)cos(σ) ) ]
= ½ θ - ½ ( sin(τ)cos(τ) - sin(σ)cos(σ) ).
Likewise
B = ½ θ - ½ ( sin(σ)cos(σ) - sin(τ)cos(τ) ).
Summing then gives
A + B = θ = length(s)
as required.
Not as elegant - but I believe also reachable by a young audience.
I'll keep an eye out for your content - but won't subscribe until you get the audio fixed up.
Good stuff. I call this base 1 math. Because of exponentiation, it wouldn't reduce to Theta if the radius were any larger. (But it would be 2x) But this same principle is how calculus and a circle's area works. The curve determines its area. It's also why Heron's Formula doesn't work that well outside of Base 1. But a+b+c=a*b*c is always a triangle. No matter how you calculate it, but the most elegant nuances are found in base 1.
One thing is still disturbing me, that is, wether this equation is dimensionally equivalent. Because one side is area and the other side is (arc) length.
7:32 The constant is the r² that is simplified here
Can you share video code??
amazing
🔥🔥🔥💯🥂🥂
thanks, I liked it.
Cool
Honestly great video, I remember struggling on this problem before giving up after a few hours.
About the mic quality. I think your setup is likely fine, just needs two things. One, move away from the mic by probably ~6 inches. Second, dont direct your mouth directly at the mic (this is fine with some, but probably not with yours), the main 'gust of air' coming from your mouth should pass just above the mic. This will prevent the sort of super intense sound you get with "p" or "k" sounds.
Good luck and cheers!
Nice
A more straight forward approach would be to simply find each of the areas using integration and adding them. take coordinates of the points as (rcosalpha,rsinalpha) and (rcosbeta,rsinbeta) and the curve is x^2 +y^2 =r^2.
Integrating and adding gives
r^2 (beta-alpha)
=r*arc length
cool vid, pls fix your mic plsplsplspls