Hey it’s Professor Dave, I wanna tell you about trigonometry.
Congratulations, we’ve made it past geometry and algebra, which means we are ready for
trigonometry.
This is a branch of mathematics that deals with triangles and angles, and it will introduce
some new functions that will seem abstract at first, but if you can wrap your head around
why we use them, they will start to make a lot of sense.
The first thing we want to do is recall what we know about angles.
Here are two coterminal rays.
This means they start at the same point and then go on indefinitely in different directions.
Between them there is an angle.
To describe this angle, we often use the coordinate plane.
We place one of the rays on the origin, extending along the X axis in the positive direction.
The location of the other ray will determine the magnitude of this angle, represented by
the Greek letter theta.
If this ray came about from a counter-clockwise rotation, we have a positive angle.
If clockwise, it’s a negative angle.
Up until now, we have been measuring angles in degrees, and we know that there are three
hundred sixty degrees in a circle, which is what we trace as we go all the way around
the coordinate plane.
Why three sixty?
Well we aren’t positive, but it seems like the ancient Babylonians were approximating
a year as three hundred sixty days.
Also, they used a counting system based on sixty, which you multiply by six to get three
sixty, and three sixty has tons of factors, so it may have been a combination of observation
and aesthetics.
Whatever the case may be, three hundred sixty degrees is a full revolution.
But now it’s time to learn another way to measure angles.
Instead of degrees we can use radians.
But what’s a radian?
Well if we have a circle of radius R, and we take two radii to form a sector whereby
this arc that connects the endpoints of the radii also has a length of R, equal to the
radii, the angle between the radii will be one radian.
Since the entire circumference of the circle is equal to pi times the diameter, or two
pi R, then going all the way around the circle must give us two pi radians, since this one
radian is equal to a radius, and two pi of these will get us all the way around the circumference.
Two pi radians is therefore equal to three hundred sixty degrees.
Pi radians brings us halfway around, or one eighty.
Half pi brings us up here to ninety, and quarter pi brings us to forty-five.
In general, to convert between degrees and radians, we can take a value in degrees, divide
by one eighty, and multiply by pi, or we could take a value in radians, divide by pi, and
multiply by one eighty.
Try converting thirty degrees to radians.
Thirty over one eighty simplifies to one over six, and then times pi, so one sixth pi.
How about two thirds pi?
Take away the pi and multiply by one eighty, that will be a hundred and twenty degrees.
We can also describe angles greater than two pi radians, or simply two pi, as we will often
take the word radian to be implied.
Say we have five pi.
Well two pi means we go around once, another two pi takes us around again, and then one
more pi puts here.
So in a sense, pi and five pi are the same angle.
We could also notice that something like five fourths pi and negative three fourths pi are
essentially the same angle as well.
Whenever we have two different angles that end up at the same spot, these are called
coterminal angles, and they will always differ by a multiple of two pi radians, or three
hundred sixty degrees.
Definitely make sure you understand this tutorial before moving on with trigonometry, as a comprehension
of what radians are and the ability to locate angles in radians on the coordinate plane
must become very easy to do.
Don’t worry, you’ll eventually memorize most of them through repetition.
In fact, this circle depicts all the angles that are multiplies of thirty degrees, or
one sixth pi, and forty five degrees, or a quarter pi.
These are the angles that we will use a lot, and they are listed in both degrees and radians.
If you are the type to memorize, I highly recommend setting your efforts on memorizing
this, as it will make things a lot easier once we start looking at trigonometric functions.
Let’s check comprehension.