Video Transcript
In this video, we’re going to learn
how to identify, measure, and describe directed angles and find their equivalent
angles.
Before we get started, though,
we’re going to recall just a couple of facts. The first is the rule for measuring
angles about a point. Angles around a point sum to 360
degrees. In other words, a full turn makes
up 360 degrees. We might also be most used to
measuring angles using degrees, such as a full turn as we’ve seen is 360. Whilst it’s not a prerequisite to
accessing this video, it’s also worth noting that we can measure angles using
radians, where a full turn, 360 degrees, is equal to two 𝜋 radians. Divide by two, and we find that 𝜋
radians is equal to 180 degrees.
But this video is about directed
angles. So, what does that mean? Well, quite simply put, a directed
angle is one which has a direction. If the angle is measured in a
counterclockwise direction, we say that angle is positive. And if it’s measured in a clockwise
direction, the angle is considered to be negative. For instance, let’s take rays 𝑂𝐴
and 𝑂𝐵. We see they have the same
origin. The directed angle here between the
rays 𝑂𝐴 and 𝑂𝐵 is positive, and this is because we’ve measured it in a
counterclockwise direction. If we measure this in a clockwise
direction, as shown, the angle is considered to be negative. 𝑂𝐴 is said to be the initial side
of the angle; in other words, it’s where we start. And then, 𝑂𝐵 is the terminal
side; that’s where the angle sort of ends.
Now, since we can continue moving
in either direction for as long as we want, it follows that there are infinitely
many determinations of the same angle. Let’s see what we actually mean by
that.
Find the smallest positive
equivalent of 788 degrees.
Let’s imagine the directed angle
788 degrees is the angle between two rays 𝑂𝐴 and 𝑂𝐵. 𝑂𝐴 is the initial side. That’s where we’re going to start
measuring the angle. And 𝑂𝐵 is the terminal side. Now, 788 degrees is positive, so
we’re going to measure the angle between these two rays in a counterclockwise
direction. We know a full turn is equal to 360
degrees. So, it follows that since 788 is
greater than 360, we’re going to need to do at least one full turn to get to this
angle. 360 degrees in a counterclockwise
direction takes us back to the ray 𝑂𝐴.
Let’s see what happens when we
complete another full turn. That’s another 360 degrees. 360 plus 360 is 720. But of course, we want to get to
788. So, let’s find the difference
between 788 and 720. That will tell us how much further
we need to go. 788 minus 720 is 68. And so, we need to travel a further
68 degrees in the counterclockwise direction. The smallest positive equivalent of
788 degrees, then, is positive 68 degrees.
We’ll now perform a similar
example, this time looking for positive equivalents of a negative angle.
Find the smallest positive
equivalent of negative 40 degrees.
Let’s imagine the directed angle
negative 40 degrees is the angle between rays 𝑂𝐴 and 𝑂𝐵. We know that a negative angle
indicates to us that we’ve measured from the initial side 𝑂𝐴 to the terminal side
𝑂𝐵 in a clockwise direction. And so, the angle, the 40 degrees,
will look a little something like this. We want to find the smallest
positive equivalent of negative 40 degrees. And so, we’re going to measure this
very same angle, but in the other direction.
The positive-directed angle
indicates to us that we measure in a counterclockwise direction. So, in other words, we’re looking
for this angle here. To find the value of this angle, we
recall that angles around a point sum to 360 degrees. And so, a positive equivalent of
negative 40 degrees is found by subtracting 40 from 360. 360 minus 40 is 320 degrees. So, this angle here is 320. The smallest positive equivalent of
negative 40 degrees, then, is 320 degrees.
Now, actually, there is a special
name for the directed angles 320 degrees and negative 40 degrees. They’re called coterminal. These are angles that share the
same initial side and same terminal sides. And so, this is a really important
definition. Coterminal angles share the same
initial and terminal sides. But of course, we’ve also seen that
we can find a coterminal angle by adding or subtracting multiples of 360 degrees or
two 𝜋 radians, depending on whether the original angle is in degrees or
radians.
We’ll now have a look at an example
of finding a positive and negative coterminal angle when the original angle is given
in radians.
Find one angle with positive
measure and one angle with negative measure which are coterminal to an angle with
measure two 𝜋 by three.
Let’s take our angle with measure
two 𝜋 by three radians. Now, if you’re not usually
confident with radians, don’t worry too much. 𝜋 radians is 180 degrees; it’s
half a turn. So, two 𝜋 by three is two-thirds
of this. Let’s draw the initial side of this
angle. And since it’s positive, we’re
going to measure in a counterclockwise direction. And so, the terminal side is likely
to be somewhere around here. Next, we recall what it means for
two angles to be coterminal. Coterminal angles are angles that
share the same initial and terminal sides.
So, essentially, we want to find
alternative ways to express the exact same angle. If we want to find, then, an angle
with positive measure, we’re going to need to keep going in the counterclockwise
direction. In fact, we’re going to need to
complete a full turn to get back to this terminal side, where a full turn is equal
to two 𝜋 radians. So, the positive coterminal angle
will be the equivalent to adding two 𝜋 by three radians and two 𝜋 radians. We can rewrite, of course, two 𝜋
as six 𝜋 over three. And the purpose of doing this is to
ensure we have a pair of equivalent denominators. And once we do, we can add the
numerators. Two plus six is eight. So, two 𝜋 by three plus six 𝜋 by
three is eight 𝜋 by three radians. And so, we have our angle with
positive measure.
What about the angle with negative
measure? Well, an angle with negative
measure is measured in a clockwise direction. We start at the same initial side,
but we travel in the opposite direction to get to the terminal side. Since the full turn is two 𝜋
radians, to find the size of this angle, we’re going to subtract two 𝜋 by three
from two 𝜋. Once again, if we rewrite two 𝜋 as
six 𝜋 by three, we can then subtract the numerators. Six 𝜋 minus two 𝜋 is four 𝜋. And so, the magnitude of the
negative angle will be four 𝜋 by three. And we, therefore, say that our two
angles are eight 𝜋 by three and negative four 𝜋 by three.
Now, we represented these with a
diagram, which really helped us figure out what was going on. But remember, when we defined a
coterminal angle, we said that we can simply add or subtract 360 degrees or two 𝜋
to our original angle to find coterminal angles. And so, an alternative way to have
found the negative angle would have been to have subtracted two 𝜋 from two 𝜋 by
three. Either way is completely acceptable
as long as we make sure to represent our angle as a negative at the end.
We’re now going to move on to
another definition. The definition we’re interested in
is that of the principal angle. We’ve already seen that there are
infinitely many determinations of the very same angle. The principal angle is the
counterclockwise angle between the initial side and terminal side that has a value
in the closed interval from zero to 360, if we’re interested in degrees, or zero
radians and two 𝜋 radians. In other words, if 𝜃 is our
terminal angle, 𝜃 can be greater than or equal to zero degrees and less than or
equal to 360 degrees or 𝜃 can be greater than or equal to zero and less than or
equal to two 𝜋. Let’s look at an application of
this idea.
Given the angle 273𝜋 over three,
find the principal angle.
We know that the principal angle is
measured in a counterclockwise direction between the initial side and terminal side
and it must have a value between zero and two 𝜋 radians. So, our job is to find the
coterminal angle to 273𝜋 over three which has a positive measure and lies in that
interval. Before we go any further, let’s see
if we can simplify this fraction somewhat. 273 divided by three is 91. So, the angle is equivalent to 91𝜋
radians. Now, we, of course, know that a
full turn is equal to two 𝜋 radians. So, essentially, we need to ask
ourselves, how many full turns can we make?
To find out, we’re going to divide
91𝜋 by two 𝜋. When we do, we simplify the
fraction by dividing both the numerator and denominator by 𝜋. And then, 91 divided by two is
45.5. In other words, we can make 45 full
turns plus another 0.5 of a turn or half of a turn. But of course, half of a turn is 𝜋
radians. And so, the principal angle has to
be 𝜋 radians.
In our final example, we’ll look at
this idea with relation to a negative angle.
Given the angle negative 23𝜋 over
five, find the principal angle.
We know that the principal angle is
the positive angle, so it’s measured in a counterclockwise direction and it has a
value in the closed interval from zero to two 𝜋 radians. And so, our job is to find the
coterminal angle to negative 23𝜋 over five which has a positive measure and lies in
this interval. So, let’s ask ourselves, what does
negative 23𝜋 over five radians actually look like? It’s negative, so it’s going to be
measured in a clockwise direction. And 23 over five is equivalent to
four and three-fifths. And we know that a full turn is two
𝜋 radians. So, we’re going to complete two
lots of full turns and another three-fifths 𝜋 radians.
So, here’s one full turn for two 𝜋
radians. Then, we complete a second full
turn, and that takes us to four 𝜋 radians. And then, we have three-fifths,
which is a little bit over one-half. And so, an angle that measures
three-fifths 𝜋 radians will look a little something like this. Now, of course, three-fifths 𝜋 is
between zero and two 𝜋. But because we’re measuring in a
clockwise direction, it’s actually negative.
To find the angle which is
coterminal to this and positive, we’re going to measure from the initial side to the
terminal side in a counterclockwise direction like this. And so, the size of this angle is
found by subtracting three 𝜋 over five from two 𝜋. By writing these numbers with the
same denominator, we could write this as 10𝜋 over five and then subtract their
numerators to get seven 𝜋 over five. And so, given an angle of negative
23𝜋 over five, the principal angle is seven 𝜋 over five.
We’re now going to recap the key
points from this lesson. In this lesson, we’ve learned that
a directed angle is one that’s given a direction. An angle measured in a
counterclockwise direction is said to be positive, whilst an angle measured in a
clockwise direction is negative. We learned that there were
infinitely many determinations of the same angle, and these are called
coterminal. Coterminal angles share the same
initial side and the same terminal side. And finally, this led us to the
definition of the principal angle. That’s the counterclockwise angle
between the initial side and terminal side. And it has a value in the closed
interval zero degrees to 360 degrees or zero radians to two 𝜋 radians.
