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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.
Start Solution

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,

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