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Section 1.3 : Trig Functions
11. Determine the exact value of sec(29π4) 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.
First we can notice that 5π4+6π=29π4 and recalling that 6π is three complete revolutions we can see that 29π4 and 5π4 represent the same angle. Next, note that π+π4=5π4 and so the line representing 5π4 will form an angle of π4 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 29π4 to the coordinates of the line representing π4 and the recall how secant is defined in terms of cosine to answer the question.
The line representing 95π4 is a mirror image of the line representing π4 and so the coordinates for 29π4 will be the same as the coordinates for π4 except that both coordinates will now be negative. So, our new coordinates will then be (−√22,−√22) and so the answer is,
sec(29π4)=1cos(29π4)=1−√2/2=−2√2=−√2