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### Section 1.3 : Trig Functions

6. Determine the exact value of $$\displaystyle \sec \left( { - \frac{{11\pi }}{6}} \right)$$ without using a calculator.

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Hint : Even though a unit circle only tells us information about sine and cosine it is still useful for secant so 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 $$\frac{\pi }{6} - 2\pi = - \frac{{11\pi }}{6}$$ and so (remembering that negative angles are rotated clockwise) we can see that the terminal line for $$- \frac{{11\pi }}{6}$$ will form an angle of $$\frac{\pi }{6}$$ with the positive $$x$$-axis in the first quadrant. In other words, $$- \frac{{11\pi }}{6}$$ and $$\frac{\pi }{6}$$ represent the same angle. So, we’ll have the following unit circle for this problem. Hint : Given the obvious symmetry here use the definition of secant in terms of cosine to write down the solution.
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Because the two angles $$- \frac{{11\pi }}{6}$$ and $$\frac{\pi }{6}$$ have the same coordinates the answer is,

$\sec \left( -\frac{11\pi }{6} \right)=\frac{1}{\cos \left( -\frac{11\pi }{6} \right)}=\frac{1}{{}^{\sqrt{3}}/{}_{2}}=\frac{2}{\sqrt{3}}$