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In the previous video, we saw that there are two systems for measuring angles.
Using the system of degree measurement, the total angular measure of a full circle is
360 degrees.
Using the system of radian measurement, the total angular measure of a full circle is
2 pi.
You will need to become fluent in both systems.
We will start this video by doing some practice with those measurements.
If going around a full circle gives a measurement of 360 degrees or 2 pi radians, then going
halfway around the circle gives us a measurement of half those quantities.
In other words, 180 degrees or pi radians.
If we divide that in half we will get a quarter of a circle, which will measure 90 degrees
or pi/2 radians.
If we divide this in half again we get 1/8 of a circle, which measures 45 degrees or
pi/4 radians.
If we return to the quarter circle and cut it into thirds we will have 1/12 of a circle.
This measures 30 degrees or pi/6 radians.
The basic angles of 30 and 45 degrees, or pi/6 and pi/4 radians, can be used to construct
the most important angles that we will encounter in this class by working our way back around
the circle using multiples of these values.
If we start with a 30-degree or pi/6 angle we can work our way around the circle to get
these measurements.
If we start with a 45 degree or pi/4 angle we get these measurements.
If we merge these into a single circle we get this diagram.
You will need to know your way around this circle.
It will become increasingly important as the semester progresses.
Initially this might seem like a lot of information.
Just remember that you can build these angles by just starting from the 30 or 45 degree
angles.
But these aren't the only angles we will encounter.
We will need a general method for switching between degrees and radians.
We will use the method of conversion factors to accomplish this.
Every equality between two quantities can be turned into a conversion factor by creating
a fraction from them, and there are two different ways to do this.
Because the numerator and the denominator are both the same value, each of these fractions
is equivalent to one.
And so we can multiply them without changing the values.
The only trick is that we must pick the one that will make the units cancel out correctly.
For example, if we're converting 105 degrees to radians we will pick the conversion factor
with degrees in the denominator so that they will cancel out.
Or if we're converting 4 pi/5 radians to degrees, we will need to pick the conversion factor
with radians in the denominator.
A word of caution regarding conversion factors:
It is easy to get in the habit that thinking the value pi must always mean radians and
that the absence of pi must always mean degrees.
You must always pay attention to the units.
For example, if we're converting 2 radians to degrees, we must still pick the conversion
factor with radians in the denominator.
It may seem strange to have a pi in your degree measurement, but remember that pi is just
a number, and you can always plug this into a calculator to get a decimal approximation
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