Let's break down the problem from the image.

Problem Recap:

You are given a point on the unit circle at $\left(-\frac{\sqrt{7}}{4}, -\frac{3}{4}\right)$ and need to find the exact values of $\sec\theta$, $\cot\theta$, and $\sin\theta$, where $\theta$ is the angle whose terminal side passes through this point.

Step 1: Determine $\sin\theta$ and $\cos\theta$

Since the point lies on the unit circle, we know: $x = \cos\theta = -\frac{\sqrt{7}}{4}, \quad y = \sin\theta = -\frac{3}{4}$

Step 2: Calculate $\sec\theta$

$\sec\theta = \frac{1}{\cos\theta} = \frac{1}{-\frac{\sqrt{7}}{4}} = -\frac{4}{\sqrt{7}} = -\frac{4\sqrt{7}}{7}$

Step 3: Calculate $\cot\theta$

$\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{-\frac{\sqrt{7}}{4}}{-\frac{3}{4}} = \frac{\sqrt{7}}{3}$

Step 4: Confirm $\sin\theta$

Given directly from the $y$-coordinate of the point: $\sin\theta = -\frac{3}{4}$

Summary of Exact Values:

$\sec\theta = -\frac{4\sqrt{7}}{7}, \quad \cot\theta = \frac{\sqrt{7}}{3}, \quad \sin\theta = -\frac{3}{4}$

Would you like to go into further detail on any of these steps or have any specific questions?

Related Questions:

1. How do you find $\cos\theta$ and $\sin\theta$ if only given $\theta$?
2. Why does the point lie on the unit circle?
3. How is $\sec\theta$ related to $\cos\theta$?
4. What is the relationship between $\cot\theta$ and $\tan\theta$?
5. How can you derive $\tan\theta$ given $\cot\theta$?

Tip:

Always ensure to rationalize the denominator when expressing values such as $\sec\theta$ or $\cot\theta$ to make them cleaner and more standardized.