State the smallest positive angle
that is equivalent to the angle shown.
Remember, when we think about
angles in standard position, the angle is measured by the amount of rotation from
the initial side to the terminal side. An angle that is measured in a
counterclockwise direction is said to be positive, whilst if it’s measured in a
clockwise direction, we say that it’s negative. We also say that if two angles in
standard position have the same terminal side, they’re called coterminal angles. So let’s examine the diagram.
We have an angle that is measured
in a clockwise direction 340 degrees. So it’s negative 340. And we’re looking to find the
measure of the coterminal angle to this, the one that lies in the first
quadrant. We can therefore use a simple angle
fact, and that is the fact that angles around a point sum to 360 degrees. The angle measured in a clockwise
direction is simply 340 degrees. It’s the absolute value of negative
340. So if we define the angle that
we’re trying to calculate to be equal to 𝜃, then we can say that 𝜃 plus 340 equals
To solve for 𝜃, we’ll simply
subtract 340 from both sides. 360 minus 340 is 20. And so the smallest positive angle
that’s equivalent to negative 340 degrees is 20 degrees.
We might also observe that the
question asked us to find the smallest positive angle. That’s because we can find an
infinite number of angles by simply completing another full turn in a
counterclockwise direction, so there will be a number of positive angles equivalent
to this that can be found by adding multiples of 360 degrees to 20.