Video Transcript
In this video, we will learn how to
identify and measure directed angles and find their equivalent angles. In order to be able to work with
directed angles, we will need to recall some key angle facts.
We begin by recalling that angles
at a point sum to 360 degrees. In other words, a full rotation
makes up 360 degrees. This can be shown on a quadrant
diagram where a quarter rotation is equal to 90 degrees and a half rotation is equal
to 180 degrees. We can also measure angles in
radians. Two 𝜋 radians equals 360
degrees. Dividing both sides by two, we see
that 𝜋 radians equals 180 degrees. This too can be shown on a quadrant
diagram.
We will now explain what we mean by
the term directed angle. A directed angle is one that has a
direction. More specifically, it can be
defined as an ordered pair of two rays, called the sides of the angle, with a common
starting point called the vertex. If an angle is measured in the
counterclockwise direction, it is said to be positive, whereas if it is measured in
the clockwise direction, it is considered to be negative.
Let us further explore exactly what
this definition means. Suppose we had the ordered pair
𝑂𝐴 and 𝑂𝐵. That is, we have two rays 𝑂𝐴, the
initial side, and 𝑂𝐵, the terminal side, and the two meet at the vertex 𝑂 as
shown. Then, we can say that 𝑂𝐴, 𝑂𝐵 is
a directed angle. Furthermore, since the angle is
measured in a counterclockwise direction, we say that it is positive. It is also possible to measure it
in a clockwise direction, which would result in a negative angle as shown.
On the other hand, if we had the
ordered pair 𝑂𝐵, 𝑂𝐴, this would correspond to the following directed angle with
two possible measures. We can see that 𝑂𝐴, 𝑂𝐵 and
𝑂𝐵, 𝑂𝐴 are not the same, since the positive and negative measures are swapped
around. In our first example, we will
consider how to find these equivalent angles measured in degrees.
Find the smallest positive
equivalent of 788 degrees.
Let’s imagine that the directed
angle 788 degrees is the angle between two rays 𝑂𝐴 and 𝑂𝐵, where 𝑂𝐴 is the
initial side and 𝑂𝐵 is the terminal side of the angle. As 788 degrees is positive, we
will measure the angle between these two rays in a counterclockwise
direction. We know that a full turn is
equal to 360 degrees. So it follows that we will have
to do at least one full turn to get the angle we require. In fact, 360 degrees plus 360
degrees equals 720 degrees, which is still less than 788 degrees. So we will need to complete two
full turns. We need to get to 788 degrees,
and 788 degrees minus 720 degrees equals 68 degrees. This means that we need to go a
further 68 degrees in the counterclockwise direction. The smallest positive
equivalent to 788 degrees is 68 degrees.
We will now consider how to find
positive equivalents when given a negative angle measured in degrees.
Find the smallest positive
equivalent of negative 40 degrees.
Let’s imagine that the directed
angle negative 40 degrees is the angle between two rays 𝑂𝐴 and 𝑂𝐵, where
𝑂𝐴 is the initial side and 𝑂𝐵 is the terminal side of the angle. As our angle is negative, we
will measure the angle between these two rays in a clockwise direction. This means that the 40-degree
angle will be as shown in the diagram.
We need to find the smallest
positive equivalent to negative 40 degrees. So we need to measure the same
angle but in the other direction. A positive directed angle
indicates we need to measure in a counterclockwise direction. We recall that angles at a
point sum to 360 degrees. So, a positive equivalent of
negative 40 degrees is found by subtracting 40 degrees from 360 degrees. Therefore, the smallest
positive equivalent of negative 40 degrees is 320 degrees.
Before looking at our next example,
we will introduce a new term, coterminal angles. In this question, negative 40
degrees and 320 degrees were examples of a pair of coterminal angles as they share
the same initial and terminal sides.
Coterminal angles share the same
initial and terminal sides. To find a coterminal angle, we can
add or subtract 360 degrees or two 𝜋 radians from the given angle. The diagrams show that 40 degrees,
400 degrees, and negative 320 degrees are coterminal angles since all three angles
have the same initial and terminal sides. Note that since we can add or
subtract as many multiples of 360 degrees as we want, there exist an infinite amount
of coterminal angles for any given directed angle.
We will now calculate 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 over three radians.
Let’s consider the angle two 𝜋
over three radians. We could convert this into
degrees by using the fact that 𝜋 radians is equal to 180 degrees. Two-thirds of 180 degrees
equals 120 degrees, so two 𝜋 over three radians equals 120 degrees. However, we will keep our angle
in radians in this question. As our angle is positive, we
need to measure the angle in a counterclockwise direction from the initial side
as shown.
We recall that coterminal
angles share the same initial and terminal sides. This means that we need to find
alternative ways to express the same angle. We can calculate coterminal
angles by adding or subtracting two 𝜋 radians from the given angle. To find another positive angle,
we need to keep measuring in the counterclockwise direction. This means that we need to add
two 𝜋 radians to our angle. Two 𝜋 radians is equivalent to
six 𝜋 over three radians. This means we need to add two
𝜋 over three and six 𝜋 over three. This is equal to eight 𝜋 over
three radians. An angle with positive measure
which is coterminal to an angle with measure two 𝜋 over three is eight 𝜋 over
three radians.
In a similar way, we can find
an angle with negative measure by moving in a clockwise direction. This means that we need to
subtract two 𝜋 from our angle. Two 𝜋 over three minus six 𝜋
over three is equal to negative four 𝜋 over three. An angle with negative measure
which is coterminal to an angle with measure two 𝜋 over three is negative four
𝜋 over three radians. The two angles are eight 𝜋
over three and negative four 𝜋 over three.
In our first three examples, we saw
that there are an infinite number of ways to describe a given angle. There will be times where we want
to restrict the size of the angle we give. When this is the case, we use the
principal angle.
The principal angle is the angle
between the initial side and the terminal side, measured in a counterclockwise
direction, that has an angle whose value in degrees lies in the closed interval from
zero to 360 and in radians lies in the closed interval from zero to two 𝜋. If 𝜃 is our principal angle, then
𝜃 is greater than or equal to zero degrees and less than or equal to 360 degrees or
greater than or equal to zero and less than or equal to two 𝜋 radians.
In our final example, we will
demonstrate how to calculate the principal angle in radians.
Given the angle negative 23𝜋
over five, find the principal angle.
The principal angle is the
counterclockwise angle between the initial side and the terminal side that has a
value 𝜃 radians, where 𝜃 is greater than or equal to zero and less than or
equal to two 𝜋 radians. This means that we need to find
the coterminal angle to negative 23𝜋 over five radians that lies between zero
and two 𝜋 radians. As the given angle is negative,
it will be measured in a clockwise direction. Since 23𝜋 over five is equal
to four and three-fifths 𝜋 and a full rotation is equal to two 𝜋 radians, we
can complete two full rotations. We then continue a further
three-fifths 𝜋 radians in the clockwise direction as shown in the diagram. Since we are measuring in a
clockwise direction, this angle is negative. It is negative three 𝜋 over
five radians.
The principal angle must be
positive. So we need to find the
coterminal angle to this measured in the counterclockwise direction. To find the coterminal angle we
need, and hence the principal angle, we subtract three 𝜋 over five from two
𝜋. This is equivalent to 10𝜋 over
five minus three 𝜋 over five, which equals seven 𝜋 over five radians. Therefore, the principal angle
of negative 23𝜋 over five is seven 𝜋 over five radians.
We will finish this video by
recapping some of the key points. A directed angle is an angle that
has a direction; the counterclockwise direction is positive and the clockwise
direction is negative. Coterminal angles share the same
initial and terminal sides. There are an infinite number of
equivalent coterminal angles. The principal angle is the
counterclockwise angle between the initial side and the terminal side that has a
value 𝜃, where 𝜃 is greater than or equal to zero degrees and less than or equal
to 360 degrees or greater than or equal to zero and less than or equal to two 𝜋
radians.
