Section 1.3 : Trig Functions
1. Determine the exact value of \(\displaystyle \cos \left( {\frac{{5\pi }}{6}} \right)\) 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 \(\pi - \frac{\pi }{6} = \frac{{5\pi }}{6}\) and so the terminal line for \(\frac{{5\pi }}{6}\) will form an angle of \(\frac{\pi }{6}\) with the negative \(x\)-axis in the second 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 \(\frac{{5\pi }}{6}\) to the coordinates of the line representing \(\frac{\pi }{6}\) and use those to answer the question.
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The coordinates of the line representing \(\frac{{5\pi }}{6}\) will be the same as the coordinates of the line representing \(\frac{\pi }{6}\) except that the \(x\) coordinate will now be negative. So, our new coordinates will then be \(\left( { - \frac{{\sqrt 3 }}{2},\frac{1}{2}} \right)\) and so the answer is,
\[\cos \left( {\frac{{5\pi }}{6}} \right) = - \frac{{\sqrt 3 }}{2}\]