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use the unit circle to evaluate each function

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

Would you like more details on any part of this solution?

### 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!

Use the unit circle to evaluate csc(7π/4)

Learn how to evaluate csc(7π/4) using the unit circle with a detailed step-by-step explanation. This solution covers the use of reference angles and the unit circle to find the value of the cosecant function, suitable for students in Grades 10-12.

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