Tom and Jana were tasked with finding the principal measure of the oriented angle: $$-\frac{20}{3}\pi$$ Each of them solved the problem differently.
Tom’s solution involved the following steps:
(1) Tom claimed that coterminal angles with the same principal measure differ by $k\cdot2\pi$, where $k\in\mathbb{Z}$.
(2) He chose $k = 3$ to find the corresponding angle with a measure close to zero: $$-\frac{20}{3}\pi+3\cdot2\pi=-\frac23\pi$$
(3) Finally, he argued that the principal measure of an oriented angle should be a positive number. Therefore, he considered the absolute value of the calculated angle as the result, which is $\frac23\pi$.
Jana’s solution involved the following steps:
(1) Jana agreed with Tom that oriented angles with the same principal measure differ by $k\cdot2\pi$, where $k\in\mathbb{Z}$.
(2) Unlike Tom, Jana chose $k = 4$, obtaining a positive angle value: $$-\frac{20}{3}\pi+4\cdot2\pi=\frac{4}{3}\pi$$
(3) Further, Jana assumed that $\frac43\pi$ corresponds to a non-convex angle and therefore considered the value of the complementary angle as the result, which is: $$2\pi-\frac43\pi=\frac23\pi.$$ (Hint: The complementary angle complements the given angle to form a full angle.)
Look carefully at the solutions of both classmates and decide which of the following statements is true:
Both Tom and Jana solved the problem correctly.
Tom solved the problem correctly. Jana made a mistake in step (3). She got the same result as Tom by chance.
Jana solved the example correctly. Tom made a mistake in step (3). He got the same result as Jana by chance.
Both Tom and Jana made a mistake in step (3). The result is correct, but they got the result by chance.
Both Tom and Jana made a mistake in step (3). The result is incorrect.
Jana's approach leads to the result more directly. Since the principal measure of an oriented angle lies within the interval $[0,2\pi)$, the angle $\frac43\pi$ calculated in step (2) of Jana's solution correctly represents the sought principal measure of the angle $-\frac{20}{3}\pi$.