What are radians? (Clear explanation)

This video gives a clear definition of radians using a steady step-by-step approach. This is part of the A Level Pure Maths curriculum in the UK and is essential for higher level maths study. #ALevel #PureMaths #Tutor #radians
Transcript:
In this video, I am going to teach you about radians by firstly drawing a circle on a set of x y axes with the centre of the circle at the origin.
This means that if we draw any line from the centre, such as the red line here that runs along the x-axis, then the length of the line will be the radius of the circle, which we call r.
This is true regardless of where we draw the line, so if we draw another line here it will also have a length equal to the radius of the circle, so again we call this length r.
Now, to give you a definition of a radian, we are going to focus on the angle created where our two lines meet, and we will call this angle theta.
So to define a radian we are going to consider the situation created when the length of the arc between our two lines also has length r. Such as that shown here.
When this is true, the three lines are all equal in length and we say that our angle theta is equal to one radian.
So one radian is simply the angle created here in the sector of a circle when the length of each side is equal to the radius.
Now, it is useful to consider how many radians fit inside the circle if we move around in an anti-clockwise direction like this.
To calculate this we are going to consider the equation for calculating the circumference of a circle which is pi times the diameter.
Since the diameter has length two times r, we can also write the length of the circumference as two pi r.
So if we divide two pi r by r (because this is the length of one radian) we find that we can cancel r and are left with two pi.
This means that the number of radians that will fit inside the circle as we move around anti-clockwise is equal to two times pi. This is approximately 6.28, which means that six full radians fit inside the circle plus a little bit extra.
So we have shown that there are two pi radians as we travel around the full circle.
This means there are pi radians in half of the circle, so we can label our axis like this.
So, if we travelled one quarter of a turn around the circle we would have pi divided by two radians.
And three quarters of a turn would be three pi divided by two, which is one and a half pi.
It is convention measure the angles in radians as we move anti-clockwise around the circle.
However, if we moved clockwise we could say that this is minus pi over 2 radians.
Or that this is minus pi radians.
So we can indicate how we are measuring our angle by using a positive or negative sign.
It will help you if you learn the common angles in radians since these will crop up frequently at advanced level and beyond.
For example, this angle (which is 45 degrees) is pi over four radians
And if we go the other way it is minus pi over four radians.
Please get in touch via the comments section if you need help with these calculations, and also press “like” if you have found this video useful. This will help it to reach others.
Thank you.

[@DavidWrightUK]

If you would like to support my channel you can now buy me a coffee here: www.buymeacoffee.com/DavidWrightUK ❤ 🙏🏼 😎

[@miraclesamuel2235]

more explanation

[@DavidWrightUK]

Thanks 👍