Many people wonder why radians do not appear when we have radians*meters. Here is an attempt at an explanation: Let s denote the length of an arc of a circle whose radius measures r. If the arc subtends an angle measuring β = n°, we can pose a rule of three: 360° _______ 2 • 𝜋 • r n° _______ s Then s = (n° / 360°) • 2 • 𝜋 • r If β = 180° (which means that n = 180, the number of degrees), then s = (180° / 360°) • 2 • 𝜋 • r The units "degrees" cancel out and the result is s = (1 / 2) • 2 • 𝜋 • r s = 𝜋 • r that is, half of the circumference 2 • 𝜋 • r. If the arc subtends an angle measuring β = θ rad, we can pose a rule of three: 2 • 𝜋 rad _______ 2 • 𝜋 • r θ rad _______ s Then s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r If β = 𝜋 rad (which means that θ = 𝜋, the number of radians), then s = (𝜋 rad / 2 • 𝜋 rad) • 2 • 𝜋 • r The units "radians" cancel out and the result is s = (1 / 2) • 2 • 𝜋 • r s = 𝜋 • r that is, half of the circumference 2 • 𝜋 • r. If we take the formula with the angles measured in radians, we can simplify s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r s = θ • r where θ denotes the "number of radians" (it does not have the unit "rad"). θ = β / (1 rad) and θ is a dimensionless variable [rad/rad = 1]. However, many consider θ to denote the measure of the angle and for the example believe that θ = 𝜋 rad and radians*meter results in meters rad • m = m since, according to them, the radian is a dimensionless unit. This solves the problem of units for them and, as it has served them for a long time, they see no need to change it. But the truth is that the solution is simpler, what they have to take into account is the meaning of the variables that appear in the formulas, i.e. θ is just the number of radians without the unit rad. Mathematics and Physics textbooks state that s = θ • r and then θ = s / r It seems that this formula led to the error of believing that 1 rad = 1 m/m = 1 and that the radian is a dimensionless derived unit as it appears in the International System of Units (SI), when in reality θ = 1 m/m = 1 and knowing θ = 1, the angle measures β = 1 rad. In the formula s = θ • r the variable θ is a dimensionless variable, it is a number without units, it is the number of radians. When confusing what θ represents in the formula, some mistakes are made in Physics in the units of certain quantities, such as angular speed. My guess is that actually the angular speed ω is not measured in rad/s but in (rad/rad)/s = 1/s = s^(-1).
I derived the area of sector in another way: For a unit circle,the area of circle or sector is proportional to its arc length I.e Area of sector/Area of circle=theta/2π multiply the Nr. and Dr. by r^2/2. Hence we get the area of sector=1/2theta×r^2.
Love your classes, mostly for refreshers, so I can tutor others. But on this one, it’s frustrating that you don’t explain exactly what a ‘radian’ is,…. In that it is the angular distance of the Radius length projected along the Circumference (57.29 degrees). That is the fundamentals of where pi (3.14159…) comes from. I think it would provide a much clearer picture if students understood this fundamental of why we use this number pi when we’re trying to explain ‘radians’. Keep up the good work!
Jason, when I first watched you in 2020, I was studying business administration and economics. After watching your videos, and seeing that I actually understand math, I now study machine learning and math at university. Thank you for changing the trajectory of my life so much. Sometimes all it takes is one video! 😊
Triva / Word Problem for ya: A Shadow moves at 15 degrees per hour. (give ro take a litttle) Using the math from the video create the formula / requirements for a sundial or a 'Math-Henge' to track the Solstice / Equinox Note: I couldn't do it. ☻
![Angular Speed & Rotational Speed in Circular Motion - [2-21-3]](http://i.ytimg.com/vi/rtXPyxVMBmM/mqdefault.jpg)
![Angular Speed & Rotational Speed in Circular Motion - [2-21-3]](/img/tr.png)
This video really improved my knowledge about area of a sector rads and others sending in confirmation representing Zambia 🇿🇲
I learnt a lot about math in this lesson Sir Jason!I think your mission is Accomplished 🎉
Awesome!
Many people wonder why radians do not appear when we have radians*meters.
Here is an attempt at an explanation:
Let s denote the length of an arc of a circle whose radius measures r.
If the arc subtends an angle measuring β = n°, we can pose a rule of three:
360° _______ 2 • 𝜋 • r
n° _______ s
Then
s = (n° / 360°) • 2 • 𝜋 • r
If β = 180° (which means that n = 180, the number of degrees), then
s = (180° / 360°) • 2 • 𝜋 • r
The units "degrees" cancel out and the result is
s = (1 / 2) • 2 • 𝜋 • r
s = 𝜋 • r
that is, half of the circumference 2 • 𝜋 • r.
If the arc subtends an angle measuring β = θ rad, we can pose a rule of three:
2 • 𝜋 rad _______ 2 • 𝜋 • r
θ rad _______ s
Then
s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r
If β = 𝜋 rad (which means that θ = 𝜋, the number of radians), then
s = (𝜋 rad / 2 • 𝜋 rad) • 2 • 𝜋 • r
The units "radians" cancel out and the result is
s = (1 / 2) • 2 • 𝜋 • r
s = 𝜋 • r
that is, half of the circumference 2 • 𝜋 • r.
If we take the formula with the angles measured in radians, we can simplify
s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r
s = θ • r
where θ denotes the "number of radians" (it does not have the unit "rad").
θ = β / (1 rad)
and θ is a dimensionless variable [rad/rad = 1].
However, many consider θ to denote the measure of the angle and for the example believe that
θ = 𝜋 rad
and radians*meter results in meters
rad • m = m
since, according to them, the radian is a dimensionless unit. This solves the problem of units for them and, as it has served them for a long time, they see no need to change it. But the truth is that the solution is simpler, what they have to take into account is the meaning of the variables that appear in the formulas, i.e. θ is just the number of radians without the unit rad.
Mathematics and Physics textbooks state that
s = θ • r
and then
θ = s / r
It seems that this formula led to the error of believing that
1 rad = 1 m/m = 1
and that the radian is a dimensionless derived unit as it appears in the International System of Units (SI), when in reality
θ = 1 m/m = 1
and knowing θ = 1, the angle measures β = 1 rad.
In the formula
s = θ • r
the variable θ is a dimensionless variable, it is a number without units, it is the number of radians.
When confusing what θ represents in the formula, some mistakes are made in Physics in the units of certain quantities, such as angular speed.
My guess is that actually the angular speed ω is not measured in rad/s but in
(rad/rad)/s = 1/s = s^(-1).
Yes, thank you. Love your teaching. Learnt soooo much.
Please, keep up these wonderful videos.❤
I derived the area of sector in another way:
For a unit circle,the area of circle or sector is proportional to its arc length I.e Area of sector/Area of circle=theta/2π multiply the Nr. and Dr. by r^2/2. Hence we
get the area of sector=1/2theta×r^2.
Love your classes, mostly for refreshers, so I can tutor others. But on this one, it’s frustrating that you don’t explain exactly what a ‘radian’ is,…. In that it is the angular distance of the Radius length projected along the Circumference (57.29 degrees). That is the fundamentals of where pi (3.14159…) comes from. I think it would provide a much clearer picture if students understood this fundamental of why we use this number pi when we’re trying to explain ‘radians’. Keep up the good work!
Thank you, this is Great
Jason, when I first watched you in 2020, I was studying business administration and economics. After watching your videos, and seeing that I actually understand math, I now study machine learning and math at university. Thank you for changing the trajectory of my life so much. Sometimes all it takes is one video! 😊
So happy to hear this! Great job and good luck to you!
Good work sir still waiting for part two
In the formula
Acs = (1 / 2) • θ • r^2
the θ denotes the number of radians (it does not have the unit "rad").
This was really helpful. Thanks For this teacher!
" Thank you so much incredible teach you're amazing in everything"
Wow that's wonderful lesson thank you very much dear teacher keep going 💕💕💕
Thank you!
Thank you, all your videos are excellent
If I had the formulas on a cheat sheet I could do pretty well with this!
Thank you
Thanks alot ❤
Sir you will live long
I hope so, and you too!
@@MathAndScience Amen
@@MathAndScience Amen
Can i use this in mathematics??
Triva / Word Problem for ya:
A Shadow moves at 15 degrees per hour. (give ro take a litttle)
Using the math from the video create the formula / requirements for a sundial or a 'Math-Henge' to track the Solstice / Equinox
Note: I couldn't do it. ☻
It’s confusing
Thank you v much indeed Sir.
Welcome!
great!
Arc length 👍
From memory...
Theta x radius x pie / 180
Awesome!
shoutout from Sweden 🇸🇪🤪💪
I’d love to visit one day!
Born great
WOW 🥰🥰🥰
❤
🤯🤯🤯🤯🤯🤯🤯🤯🤯
WOW HOW ABOUT THE OLD {PIE ARED SQUARED DIVIDED BY 360/ TIMES DEGREES?? IF YOU CAN`T EXPLAIN TOO A 6 YEAR OLD/ YOU YA YOU DON`T UNDERSTAND
AND THE ARC LENGTH? DIA, TIMES PI. DIVIDED BY 360/ TIMES ANGLE
Ukochini
😂
That works👍
sir please talk to slwoly
Sir please know how to spell slowly-
Sir please listen faster.
first👍😎