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Section 1.3 : Trig Functions

10. Determine the exact value of \(\displaystyle \sin \left( { - \frac{{11\pi }}{3}} \right)\) 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 \(\frac{\pi }{3} - 4\pi = - \frac{{11\pi }}{3}\) and note that \(4\pi \) is two complete revolutions (also, remembering that negative angles are rotated clockwise) we can see that the terminal line for \( - \frac{{11\pi }}{3}\) and \(\frac{\pi }{3}\) are the same angle and so we’ll have the following unit circle for this problem.

Hint : Given the very obvious symmetry here write down the answer to the question.
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Because \( - \frac{{11\pi }}{3}\) and \(\frac{\pi }{3}\) are the same angle the answer is,

\[\sin \left( { - \frac{{11\pi }}{3}} \right) = \frac{{\sqrt 3 }}{2}\]