Question

watch the video and then solve the problem given below. click here to watch the video. use reference angles to find the exact value of the following expression. do not use a calculator. $csc\frac{5pi}{4}$

Explanation:

Step1: Recall the definition of cosecant

$\csc\theta=\frac{1}{\sin\theta}$, so $\csc\frac{5\pi}{4}=\frac{1}{\sin\frac{5\pi}{4}}$.

Step2: Find the reference - angle

The angle $\theta = \frac{5\pi}{4}$ is in the third - quadrant. The reference angle $\theta'$ for an angle $\theta$ in the third - quadrant is $\theta-\pi$. So, $\theta'=\frac{5\pi}{4}-\pi=\frac{\pi}{4}$.

Step3: Determine the sign of sine in the third - quadrant

In the third - quadrant, the sine function is negative. And $\sin\theta=-\sin\theta'$ for $\theta$ in the third - quadrant. Since $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, then $\sin\frac{5\pi}{4}=-\frac{\sqrt{2}}{2}$.

Step4: Calculate the value of cosecant

Since $\csc\frac{5\pi}{4}=\frac{1}{\sin\frac{5\pi}{4}}$, substituting $\sin\frac{5\pi}{4}=-\frac{\sqrt{2}}{2}$ gives $\csc\frac{5\pi}{4}=\frac{1}{-\frac{\sqrt{2}}{2}}=-\sqrt{2}$.

Answer:

$-\sqrt{2}$