To evaluate $\csc\left(\frac{7\pi}{4}\right)$ using the unit circle, let's proceed step-by-step:

Step 1: Identify the corresponding angle on the unit circle

The angle $\frac{7\pi}{4}$ is in radians. Since one full circle is $2\pi$ radians, this angle exceeds $2\pi$. However, we can subtract $2\pi$ from $\frac{7\pi}{4}$ to find an equivalent angle: $\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}$ This tells us that $\frac{7\pi}{4}$ corresponds to $-\frac{\pi}{4}$ on the unit circle, which is in the fourth quadrant.

Step 2: Find the sine of $\frac{7\pi}{4}$

In the unit circle, the reference angle for $\frac{7\pi}{4}$ is $\frac{\pi}{4}$, which has a sine value of $\frac{\sqrt{2}}{2}$. Since $\frac{7\pi}{4}$ is in the fourth quadrant, where sine is negative, we have: $\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

Step 3: Calculate the cosecant

Cosecant is the reciprocal of sine, so: $\csc\left(\frac{7\pi}{4}\right) = \frac{1}{\sin\left(\frac{7\pi}{4}\right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2}$

Thus, $\csc\left(\frac{7\pi}{4}\right) = -\sqrt{2}$.

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Related Questions:

1. What is the sine of $\frac{7\pi}{6}$?
2. How do you calculate the cosine of $\frac{5\pi}{3}$?
3. What is the tangent of $\frac{3\pi}{4}$ using the unit circle?
4. How do you evaluate $\sec\left(\frac{11\pi}{6}\right)$?
5. What is the value of $\cot\left(\frac{5\pi}{4}\right)$?

Tip: When working with the unit circle, it's helpful to memorize the sine and cosine values of key angles such as $\frac{\pi}{6}, \frac{\pi}{4},$ and $\frac{\pi}{3}$ for faster evaluations!