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Section 1.3 : Trig Functions
4. Determine the exact value of cos(−2π3) without using a calculator.
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Hint : Sketch a unit circle and relate the angle to one of the standard angles in the first quadrant.
First we can notice that −π+π3=−2π3 so (recalling that negative angles rotate clockwise and positive angles rotation counter clockwise) the terminal line for −2π3 will form an angle of π3 with the negative x-axis in the third quadrant and we’ll have the following unit circle for this problem.
Hint : Given the obvious symmetry in the unit circle relate the coordinates of the line representing −2π3 to the coordinates of the line representing π3 and use those to answer the question.
The line representing −2π3 is a mirror image of the line representing π3 and so the coordinates for −2π3 will be the same as the coordinates for π3 except that both coordinates will now be negative. So, our new coordinates will then be (−12,−√32) and so the answer is,
cos(−2π3)=−12