How do you evaluate $sec (\frac{5 π}{4})$ $sec((5pi)/4)$ ?

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How do you evaluate $sec (\frac{5 π}{4})$ $sec((5pi)/4)$ ?

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Answer 1 · 2016-03-19T21:25:57

secant is the reciprocal of COSINE

so sec $\frac{5 π}{4}$ $(5pi)/4$ = $\frac{1}{cos (\frac{5 π}{4})}$ $1/(cos((5pi)/4)$
Now the angle is in 3rd quadrant and cosine is negative in the 3rd quadrant (CAST rule).

this means that the $\frac{1}{cos (\frac{5 π}{4})}$ $1/(cos((5pi)/4)$ = $- \frac{1}{cos (\frac{π}{4})}$ $-1/(cos((pi)/4)$
and since $cos (\frac{π}{4}) = \frac{1}{\sqrt{2}}$ $cos((pi)/4)=1/sqrt2$ , your result is that

sec $\frac{5 π}{4} = - \frac{\sqrt{2}}{1}$ $(5pi)/4=-sqrt2/1$

hope this helps

Answer 2 · 2016-03-19T23:53:28

$- \sqrt{2}$ $-sqrt2$

Explanation:

$sec (\frac{5 π}{4}) = \frac{1}{cos (\frac{5 π}{4})}$ $sec ((5pi)/4) = 1/(cos ((5pi)/4))$ .
Find cos ((5pi)/4)
Trig unit circle and trig table give -->
$cos (\frac{5 π}{4}) = cos (\frac{3 π}{4}) = - \frac{\sqrt{2}}{2}$ $cos ((5pi)/4) = cos ((3pi)/4) = -sqrt2/2$
Therefor:
$sec (\frac{5 π}{4}) = - \frac{2}{\sqrt{2}} = - \sqrt{2}$ $sec ((5pi)/4) = - 2/sqrt2 = - sqrt2$

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