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sjeikdom0

Answered question

2021-09-14

Evaluate the trigonmetric function at the given real number:

\csc (\frac{5 π}{4})

Theodore Schwartz

Skilled2021-09-15Added 99 answers

\frac{3 π}{2} > \frac{5 π}{4} > π

Thus, the angle lies in third quardant.
Therfore,

\csc (\frac{5 π}{4}) = \frac{1}{\sin (\frac{5 π}{4})}

(a)

\sin (\frac{5 π}{4}) = \sin (π + \frac{π}{4})

\sin (π + \frac{π}{4}) = - \sin (\frac{π}{4})

- \sin (\frac{π}{4}) = - \frac{1}{\sqrt{2}}

(\sin (\frac{5 π}{4}) = - \frac{1}{\sqrt{2}}

Substitute the value in (a)

\csc (\frac{5 π}{4}) = - \frac{1}{\frac{1}{\sqrt{2}}}

\csc (\frac{5 π}{4}) = - \sqrt{2}

\csc (\frac{5 π}{4}) = - (\frac{\sqrt{2} \times \sqrt{2}}{\sqrt{2}})

= - \frac{2}{\sqrt{2}}

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