Question

Evaluate: $\mathrm{cosec}\frac{-3\mathrm{\pi }}{4}$.

Answer 1

Evaluating: $\cos ec\frac{-3\pi }{4}$

Adding full rotations of $2\mathrm{\pi }$ until the angle is larger than or equal to ${\displaystyle 0}$ and smaller than ${\displaystyle 2\pi }$.

Gives, $cosec\left(\frac{5\pi }{4}\right)$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

Make the expression negative because cosecant is negative in the third quadrant.

$-cosec\left(\frac{\pi }{4}\right)$

The exact value of $cosec\left(\frac{\pi }{4}\right)$ is ${\displaystyle \sqrt{2}}$.

Therefore, the value of $-cosec\left(\frac{\pi }{4}\right)$ is ${\displaystyle -\sqrt{2}}$.The result can be shown in multiple forms.

Hence, the value of $-cosec\left(\frac{\pi }{4}\right)$ is ${\displaystyle -\sqrt{2}}$.