https://file.aiquickdraw.com/mathbot/file/1727054979166isnuqtkr.jpg

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.

Suppose that θ is an angle in standard position whose terminal side intersects the unit circle at (-√7/4, -3/4). Find the exact values of sec(θ), cot(θ), and sin(θ).

Learn how to find the exact values of sec(θ), cot(θ), and sin(θ) for an angle θ whose terminal side intersects the unit circle at (-√7/4, -3/4). This solution covers trigonometric functions and their relations. Suitable for high school students in Grades 10-12.

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