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