Video Transcript
In this lesson, we’ll learn how to
identify angles in standard position and find the positive and negative measures of
their equivalent angles. We’re going to be referring to the
four quadrants throughout this lesson, so we’ll remind ourselves what we mean by
these. We split the coordinate plane into
four quarters called quadrants. We label them in a counterclockwise
direction and use Roman numerals to do so, with this quadrant being the first, this
being the second, this being the third, and this being the fourth.
We might also be used to measuring
angles using degrees, such that a full turn is 360 degrees. Now, whilst not a prerequisite to
accessing the video, it’s worth noting that we can also measure angles using radians
where a full turn, 360 degrees, is equal to two 𝜋 radians. So, this lesson is about angles in
standard position. But what does that actually
mean? An angle is said to be in standard
position if its vertex is located at the origin and one ray lies on the positive
𝑥-axis. The ray that lies on the 𝑥-axis is
called the initial side, and the other ray is called the terminal side.
We also say that if the terminal
side of an angle lies on one of the axes, such as at 90 degrees or 180 degrees, it’s
called a quadrantal angle. The angle is then measured by the
amount of rotation from the initial side to the terminal side. If the angle is measured in a
counterclockwise direction, we say that the measurement is positive. But if we measure the angle in a
clockwise direction, we say that the measurement is negative.
Our final definition refers to two
angles which are in standard position and have the same terminal side, and those are
called coterminal, so we can see from the diagram here that 60 degrees and negative
300 degrees are coterminal angles. So, let’s practice identifying
whether angles are actually in standard position.
Is the angle in standard
position?
An angle is said to be in standard
position if its vertex is located at the origin and one ray, which we call the
initial side, lies on the positive 𝑥-axis. Now, our angle is bound by the rays
highlighted in yellow. We do indeed see that one of the
sides lies on the positive 𝑥-axis and the vertex does appear to be at the
origin. We also know that it is indeed the
initial side of our angle that lies on the positive 𝑥-axis rather than the terminal
side. And this is because we’re given an
arrow that tells us the direction.
If the arrow was moving in the
opposite direction, we couldn’t actually say that the angle is in standard position
because the initial side would not lie on the positive 𝑥-axis. And so, we see that the angle given
satisfies all of our criteria to be in standard position. And the answer is yes.
Is the angle in standard
position?
Remember, we say that an angle is
in standard position if its vertex is located at the origin and one ray, which is
the initial side, lies on the positive 𝑥-axis. Our angle is the angle measured in
a counterclockwise direction between the two lines highlighted in yellow. One of these rays does indeed lie
on the positive 𝑥-axis. And in fact, we can see it’s the
initial side because we’re given the direction in which the angle is measured. We can also see that the vertex,
the point where our initial side and terminal side meet, is at the origin. And so our angle satisfies all the
relevant criteria required to be considered in standard position. And the answer is yes.
Let’s now look an example defined
by an ordered pair.
Does the ordered pair given by the
ray joining 𝐶𝐴 and the ray joining 𝐶 to 𝐷 express an angle in standard
position?
We know that for an angle to be in
standard position, its vertex must be centered at the origin and the initial side
must lie on the positive 𝑥-axis. Now, our angle is defined by this
ordered pair. So we have the ray joining 𝐶 and
𝐴 and the ray joining 𝐶 and 𝐷. The ray joining 𝐶 to 𝐴 is this
one, and that does indeed lie on the positive 𝑥-axis. We know it’s the initial side of
our angle because it’s mentioned first. Then the ray that joins 𝐶 to 𝐷 is
this one here, meaning the angle we’re interested in is this one. But does the vertex of this lie at
the origin? Well, no, the vertex is over here
somewhere. It’s someway along the positive
𝑥-axis. And so the ordered pair defined by
the ray joining 𝐶 to 𝐴 and the ray joining 𝐶 to 𝐷 does not express an angle in
standard position. And the answer is no.
There are, however, a couple of
angles that are in standard position here. The first would be given by the
ordered pair the ray joining 𝑂 to 𝐶 and the ray joining 𝑂 to 𝐸. The ray 𝑂𝐶 lies on the positive
𝑥-axis, and then the vertex is centered at the origin. Similarly, we could begin with the
same initial side, and that’s the ray joining 𝑂 to 𝐶, and measure through to the
ray joining 𝑂 to 𝐺. The vertex for this angle is still
located at the origin, and so the angle is also in standard position.
In our next example, we’ll look at
how we can find the value of coterminal angles.
State the positive related angle
for the angle shown.
Remember, when thinking about
angles in standard position, the angle is measured by the amount of rotation from
the initial side to the terminal side. If we measure our angle in a
counterclockwise direction, we consider the measurement to be positive. And if we measure in a clockwise
direction, we say that the measurement is negative. We also say that if two angles in
standard position have the same terminal side, they’re called coterminal angles. And so here we have negative 340
degrees. And of course, since this is
negative, it’s measured in a clockwise direction.
To find the size of the coterminal
angle, we need to identify where the terminal side is. The terminal side is this one. And so to find the positive related
angle, we need to find the measure of this angle here, this acute angle. And so let’s use one of the key
angle facts. We know that angles around a point
sum to 360 degrees. And so we take the measure of the
reflex angle to be 340 degrees — remember, the negative just tells us the direction
— and we’re going to subtract its value from 360. 360 minus 340 is 20 degrees. Since this angle is indeed being
measured in a counterclockwise direction from its initial side, we say that it’s
positive. And so the positive related angle
for the angle we were given is 20 degrees.
Now that we’ve seen how to find the
value of coterminal angles, let’s look at finding multiple coterminal angles.
Find a positive and a negative
coterminal angle for 340 degrees.
Remember, when we think about
angles in a standard position, the angle is measured by the amount of rotation from
the initial side to the terminal side. If we measure our angle in a
counterclockwise direction, we consider the measurement to be positive. And if we measure in a clockwise
direction, the measurement is negative. Now we also say that if two angles
in standard position have the same terminal side, then they are called coterminal
angles. So, let’s draw the angle 340
degrees out first.
The vertex of our angle needs to
lie at the origin, and its initial side must lie on the positive 𝑥-axis. So, that’s the yellow line
shown. It’s positive 340 degrees, so we’re
going to measure 340 degrees in a counterclockwise direction. A full turn, of course, is 360
degrees. So it’s a little bit less than a
full turn. And so our angle in standard
position will look a little something like this. We need to find another positive
coterminal angle and a negative coterminal angle. We’ll start with a negative
coterminal angle since it’s a little easier to work out.
Since we’re measuring from the
initial side to the terminal side but we want the measurement to be negative, we’re
going to go in a clockwise direction. That’s simply here to here. And so we’re going to subtract the
angle we were given, 340 degrees, from the amount of degrees in a full turn, that’s
360. 360 minus 340 is 20 or 20
degrees. So the angle is 20 degrees. But because we’re measuring it in a
clockwise direction, we say that it’s negative 20 degrees.
So that’s our negative coterminal
angle. But how do we find the positive
one? Well, in fact, what we’re going to
do is complete the 340-degree turn, and then we’re going to complete another full
turn. In doing so, we end up back at the
same terminal angle. And so to find the value of this
positive coterminal angle, we’re going to add a full turn onto the initial turn. So 340 plus that full turn, 360
degrees. 340 plus 360 is 700. And of course, we measure this in a
counterclockwise direction, so it’s positive 700 degrees. And so the positive and negative
coterminal angle for 340 degrees are 700 degrees and negative 20 degrees.
And, of course, it follows that we
could continue to complete turns and still end up on that terminal side. This means that there are, in fact,
an infinite number of variations for both the positive and negative coterminal
angle.
In our final example, we’ll look at
how to identify the quadrant in which an angle in standard position lies.
In which quadrant does the angle
negative 242 degrees lie?
Remember, if we take the coordinate
plane, we can split it into four quarters which are called quadrants. We label these in a
counterclockwise direction, and we use Roman numerals to do so. We say that this quadrant is the
first quadrant. We have the second over here. The third then lies down here. And this is the fourth
quadrant. So in which quadrant does the angle
negative 242 degrees lie?
Well, to find out, we’re going to
begin by adding the initial side of our angle onto the diagram. The vertex of our angle has to lie
at the origin and its initial side must be located on the positive 𝑥-axis as
shown. Since the angle is negative, we
need to travel in a clockwise direction. We’ll do this in intervals of 90
degrees. 90 degrees takes us to this line
here. Then another 90 degrees takes us
half a turn. It takes us 180 degrees. And our terminal side would lie on
the negative 𝑥-axis. If we were to continue another 90
degrees, that would take us all the way to 270 degrees.
And in fact, of course, since we’re
traveling in a clockwise direction, they should all be negative. But we only wanted 242 degrees. And so we’re not going to quite
complete that last quarter of a turn. We can therefore say that the angle
negative 242 degrees lies in the second quadrant.
Let’s recap the key points from
this lesson. An angle is said to be in standard
position if its vertex is located at the origin and one ray is on the positive
𝑥-axis. The ray that lies on the 𝑥-axis is
called the initial side. And the other ray that forms the
angle is called the terminal side. We also define a quadrantal angle
to be made up of one in which the terminal side lies on one of the axes, such as at
270 degrees or 360 degrees. Then, the angle is measured by the
amount of rotation from the initial side to the terminal side. And if we measure in a
counterclockwise direction, that measurement is positive. If we measure in a clockwise
direction, the measurement is negative.
Finally, we said that if two angles
in standard position have the same terminal side, then we call those coterminal
angles.
