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To solve for \(\cot \theta\) given that \(\cos \theta = \frac{21}{29}\) and \(\theta\) is in quadrant IV, we can follow these steps:

### Step 1: Use the Pythagorean Identity
The Pythagorean identity states:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Given \(\cos \theta = \frac{21}{29}\), we substitute and solve for \(\sin \theta\):

\[
\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{21}{29}\right)^2
\]

\[
\sin^2 \theta = 1 - \frac{441}{841} = \frac{841 - 441}{841} = \frac{400}{841}
\]

\[
\sin \theta = \pm \frac{20}{29}
\]

### Step 2: Determine the Sign of \(\sin \theta\)
Since \(\theta\) is in quadrant IV, \(\sin \theta\) is negative. Therefore:

\[
\sin \theta = -\frac{20}{29}
\]

### Step 3: Calculate \(\cot \theta\)
The cotangent function is defined as:

\[
\cot \theta = \frac{\cos \theta}{\sin \theta}
\]

Substituting the known values:

\[
\cot \theta = \frac{\frac{21}{29}}{-\frac{20}{29}} = -\frac{21}{20}
\]

So, the value of \(\cot \theta\) is \(\boxed{-\frac{21}{20}}\).

Would you like more details or have any questions? 

Here are five related questions to expand on this topic:
1. How would the value of \(\cot \theta\) change if \(\theta\) were in quadrant II?
2. Can you derive \(\tan \theta\) using the values of \(\sin \theta\) and \(\cos \theta\)?
3. What are the possible values of \(\theta\) if \(\cos \theta = \frac{21}{29}\) in degrees?
4. How can the Pythagorean identity be used to find \(\sec \theta\)?
5. What would be the value of \(\sin(2\theta)\) using the given values?

**Tip:** When solving trigonometric problems, always remember to consider the quadrant, as it determines the sign of the sine and cosine functions.

Find \(\cot \theta\) given that \(\cos \theta = \frac{21}{29}\) and \(\theta\) is in quadrant IV.

Learn how to find \(\cot \theta\) when \(\cos \theta = \frac{21}{29}\) and \(\theta\) is in quadrant IV. This solution covers trigonometric identities, Pythagorean identity, and the determination of signs in different quadrants. Ideal for students in Grades 10-12.

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