Calculus I - Trig Functions

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Section 1.3 : Trig Functions

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1. Determine the exact value of $\displaystyle \cos \left( {\frac{{5\pi }}{6}} \right)$ without using a calculator.

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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.

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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,

$\cos \left( {\frac{{5\pi }}{6}} \right) = - \frac{{\sqrt 3 }}{2}$