Trigonometry - Finding the arc length of a circle

[ericgonzalez]

Free math videos? What a blessing, everything is well explained and once again, free! Thank you very much so helpful.

[OneTrickPony]

This man is a good human. That is all I have to say.

[JeffreyVergara]

I'm definitely gonna use this for my trig class

[JoséAntonioBottino]

At minute 0:34 you say “as long that angle is written in radians”. That is not so. The variable θ is the number of radians and not the measure of the angle.

Here is the explanation of the formula:

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

that is, half of the circumference 2 • 𝜋 • r
s = 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

that is, half of the circumference 2 • 𝜋 • r
s = 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

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 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).

[dalesmr]

THANK YOU MAN IM HOMELESS BUT NOT ANYMORE