Video Transcript
Which of the following angles is
equivalent to an angle measuring 20 degrees in standard position?
We have been given five options
measured in degrees. Option (A) is 110 degrees, option
(B) negative 70 degrees, option (C) 200 degrees, option (D) 380 degrees, and option
(E) negative 160 degrees.
First, we will recall what is meant
by an angle in standard position. An angle in standard position has
its vertex on the origin and one ray is on the positive 𝑥-axis. This allows us to sketch the
original angle measuring 20 degrees in standard position. We place the vertex of our angle on
the point zero, zero. We call the side of the angle that
is on the 𝑥-axis the initial side. 20 degrees is a positive measure,
so we rotate counterclockwise around the origin. This gives us the terminal side of
our angle.
In this question, we’re asked to
find an equivalent angle measuring 20 degrees. Considering none of the five
options are numerically equivalent to 20 degrees, what could this possibly mean? In this context, we will be able to
find infinitely many equivalent positive or negative angles. These equivalent angles in standard
position are formally referred to as coterminal angles. What makes two angles in standard
position on the coordinate plane coterminal is that they share the same terminal
side.
In order to find the measure of a
coterminal angle, we will consider the fact that one full turn around the origin
measures 360 degrees. If, after rotating the terminal
side of an angle in standard position on a coordinate plane 20 degrees
counterclockwise about the origin, we were then to rotate it 360 more degrees about
the origin in the counterclockwise direction, our terminal side would be in the same
position it was before the additional 360-degree rotation. In fact, we could repeat this
process by adding 360 degrees as many times as we would like to find more positive
coterminal angles to 20 degrees. This means that the first positive
coterminal angle to 20 degrees is 380 degrees, followed by 740 degrees, 1100
degrees, 1460 degrees, and so on.
To find negative coterminal angles,
we would simply rotate in the negative clockwise direction about the origin. This means subtracting 360 degrees
or any multiple of 360 degrees. Therefore, the first negative angle
coterminal to 20 degrees is negative 340 degrees, followed by negative 700 degrees,
negative 1060 degrees, negative 1420 degrees, and so on.
We have now listed four positive
coterminal angles and four negative coterminal angles. Only one of the five options we
were given is found on these lists, and that is 380 degrees. In conclusion, from the five
options we were given, only 380 degrees is equivalent to an angle measuring 20
degrees in standard position.
