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Section 1.3 : Trig Functions

4. Determine the exact value of \(\displaystyle \cos \left( { - \frac{{2\pi }}{3}} \right)\) 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.
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First we can notice that \( - \pi + \frac{\pi }{3} = - \frac{{2\pi }}{3}\) so (recalling that negative angles rotate clockwise and positive angles rotation counter clockwise) the terminal line for \( - \frac{{2\pi }}{3}\) will form an angle of \(\frac{\pi }{3}\) with the negative \(x\)-axis in the third quadrant and we’ll have the following unit circle for this problem.

TrigFcns_Prob4
Hint : Given the obvious symmetry in the unit circle relate the coordinates of the line representing \( - \frac{{2\pi }}{3}\) to the coordinates of the line representing \(\frac{\pi }{3}\) and use those to answer the question.
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The line representing \( - \frac{{2\pi }}{3}\) is a mirror image of the line representing \(\frac{\pi }{3}\) and so the coordinates for \( - \frac{{2\pi }}{3}\) will be the same as the coordinates for \(\frac{\pi }{3}\) except that both coordinates will now be negative. So, our new coordinates will then be \(\left( { - \frac{1}{2}, - \frac{{\sqrt 3 }}{2}} \right)\) and so the answer is,

\[\cos \left( { - \frac{{2\pi }}{3}} \right) = - \frac{1}{2}\]