The following graph shows a right triangle at 𝐵, where 𝑂𝐵 is equal to two root three and 𝐵𝐶 is equal to two. Which of the following is a coterminal angle of 𝑂? Is it (A) 60 degrees, (B) 300 degrees, (C) 330 degrees, (D) 390 degrees, or (E) 420 degrees?
Let’s begin by adding the information we are given in the question to our diagram. We are told that in the right triangle, side length 𝑂𝐵 is equal to two root three, and side length 𝐵𝐶 is equal to two. We’re asked to find the coterminal angle of 𝑂.
We recall that coterminal angles are angles in standard position that share the same terminal side. Angles in standard position are measured from the positive 𝑥-axis. And this means that the initial side of any angle in standard position must lie along this axis. The terminal side of angle 𝑂 is the hypotenuse of our right triangle, side length 𝑂𝐶.
Before trying to find any coterminal angles, we will calculate the angle at 𝑂 labeled 𝜃. We will do this using our knowledge of right angle trigonometry and the tangent ratio, which states that tan 𝜃 is equal to the opposite over the adjacent. Substituting in the values from our triangle, we have tan 𝜃 is equal to two over two root three. By canceling a common factor of two from the numerator and denominator, this simplifies to one over root three.
At this stage, we may recall that the tangent of one of our special angles, 30 degrees, is equal to one over root three. This means that 𝜃 is equal to 30 degrees. Alternatively, we could’ve taken the inverse tangent of both sides of our equation, giving us 𝜃 is equal to the inverse tan of one over root three. Ensuring we’re in degree mode, we could type this into our calculator, giving us 𝜃 is equal to 30 degrees.
We are now in a position to work out which of our options is a coterminal angle of 30 degrees. Every angle has an infinite number of positive and negative coterminal angles. These can be found by adding integer multiples of 360 degrees to the angle or subtracting them from the angle. In this case, adding 360 degrees to our value of 𝜃 gives us 390 degrees. This corresponds to option (D). So we can therefore conclude that a coterminal angle of 𝑂 is 390 degrees.
None of the other options can be found by adding or subtracting integer multiples of 360 degrees to angle 𝑂.