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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,

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