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Section 1.3 : Trig Functions

11. Determine the exact value of \(\displaystyle \sec \left( {\frac{{29\pi }}{4}} \right)\) 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 \(\frac{{5\pi }}{4} + 6\pi = \frac{{29\pi }}{4}\) and recalling that \(6\pi \) is three complete revolutions we can see that \(\frac{{29\pi }}{4}\) and \(\frac{{5\pi }}{4}\) represent the same angle. Next, note that \(\pi + \frac{\pi }{4} = \frac{{5\pi }}{4}\) and so the line representing \(\frac{{5\pi }}{4}\) will form an angle of \(\frac{\pi }{4}\) with the negative \(x\)-axis in the third 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 \(\frac{{29\pi }}{4}\) to the coordinates of the line representing \(\frac{\pi }{4}\) and the recall how secant is defined in terms of cosine to answer the question.
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The line representing \(\frac{{95\pi }}{4}\) is a mirror image of the line representing \(\frac{\pi }{4}\) and so the coordinates for \(\frac{{29\pi }}{4}\) will be the same as the coordinates for \(\frac{\pi }{4}\) except that both coordinates will now be negative. So, our new coordinates will then be \(\left( { - \frac{{\sqrt 2 }}{2}, - \frac{{\sqrt 2 }}{2}} \right)\) and so the answer is,

\[\sec \left( \frac{29\pi }{4} \right)=\frac{1}{\cos \left( \frac{29\pi }{4} \right)}=\frac{1}{-{}^{\sqrt{2}}/{}_{2}}=-\frac{2}{\sqrt{2}}=-\sqrt{2}\]