Let’s take a breather, and deviate for a while.
Let’s talk about the dracion way of measuring rotation that we’ve studied all our lives
and continued to use so far in this course.
The degree.
We all know that a circle is 360 degree.
360 seems like such an arbitrary number, it must come from somewhere, must mean something,
right?
I honestly have no idea where that number comes from, there are a couple of theories
linking the historical understanding that a year was 360 days in length, and other accounts
which link to civilizations having a base 60 number system.
But there is a more scientific, more grounded, more universal measure for an angle, the Radian
and I want to convince you to start using it, because everybody else is.
All trigonometric functions, when writing a sin function in a programming language for
instance or when type the sin function on a scientific calculator, all expect the angle
to be given in radian and not in degree.
So what is a radian and why is it universal?
You can think of a radian as a measure of the length of the circle that an angle represents.
Instead of thinking about how many degrees of rotation is needed to go from one point
on a circle to another, we think about the distance needed to be covered to get to that
point along the path of the circle.
A radian is basically a measure of the length of the arc that subtends an angle.
An angle is said to be 1 Radian if it subtends an arc of length equal to the radius of the
circle.
Alright, but why is it universal?
To understand this, we need to talk about Pi.
Whole books are dedicated to the study of pi, so keeping it brief, it's the ratio of
the circumference of the circle to its diameter and it’s a universal constant, the golden
number of 3.14159 and so on.
So if the mathematics of rotation were to refer to this constant, we know that it will
be universally valid.
The radian angle of a full circle is the length of the full circle, which is just the circumference.
And the circumference of a circle is given by the formula 2PiR, and we know that the
radius is one for a unit circle, so the radian angle of a full unit circle is 2Pi.
Half a circle, where we previously referred to as 180 is Pi radian, Similarly, a quarter
of a circle is Pi over 2.
So you can see a pattern here, the length of the radian or the radian angle is alway
expressed as a ratio of Pi.
An expression which has the value of Pi in it should really only be evaluated at the
last step of calculating the result.
The reason is that the radian is not entirely intuitivity in its calculated form.
For example, visualize a section of a circle whose angle is 1.74 radian.
Hard isn’t it.
It’s hard to even know which quadrant this angle lies in, is it lesser than a quarter
circle, more than half?
Well, in the traditional sense, it’s a 100 degrees.
Expressed as a fraction of PI, it's 5/9PI, which is still not overly intuitive, but at
least we can picture it in the 2nd quadrant since it’s over half the value of PI.
To wrap things up, we can look at how many degrees are in 1 radian, and with that a generalized
formula for converting between the two.
360 degree is 2 Pi rad, by proportional equality, how many degrees is equal to 1 radian.
Solving for this equation, we get the value of 1 radian to be approximately 57.29 degrees.
And the general formula for converting between the 2.
In the next video, we’ll be resuming the discussion on sin functions, but now, as a
function of time.