Find the first negative angle which
is coterminal with 190 degrees.
We′ll come back to what the word
coterminal means in a moment. But first, let′s consider what an
angle of 190 degrees looks like in standard position.
When sketching an angle in standard
position, the initial side of the angle lies along the positive 𝑥-axis. Convention when describing angles
on a coordinate system is that positive angles are measured in a counterclockwise
direction from the positive 𝑥-axis, whereas negative angles are measured in a
clockwise direction. This means that we can mark on the
positive angles from zero to 360 degrees as shown.
The angle that we′re interested in
is positive 190 degrees, which means it′s 190 degrees in a counterclockwise
direction from the positive 𝑥-axis. We can see from our diagram that
this lies in the third quadrant, as an angle of 180 degrees would be on the negative
𝑥-axis and we′re going 10 degrees further.
Now let′s consider what this
question is asking us to do. It wants us to find the first
negative angle which is coterminal with this angle of 190 degrees. We know that two angles are
coterminal if they have the same initial and terminal sides. This means that we′re looking for a
different way of describing the angle that we′ve already drawn. As we are looking for a negative
value, this means that instead of describing the angle in a counterclockwise
direction, we instead describe it in a clockwise direction.
We can add the values negative 90
degrees and negative 180 degrees to our diagram. And it is clear that the first
negative angle lies between these two values. We know that angles around a point
sum to 360 degrees, which means that we can find the magnitude of this angle by
subtracting 190 from 360, giving us 170 degrees. We then need to make this value
negative so that it describes movement in a clockwise direction. The first negative angle which is
coterminal with 190 degrees is negative 170 degrees.
It is worth noting that this
question specifically wanted the first negative angle. The reason for this is that there
are an infinite number of coterminal angles. And we can find these by adding or
subtracting 360 degrees to our angle. For example, the angles negative
530 degrees, negative 170 degrees, 190 degrees, and 550 degrees are all