To solve the equation \( \cot(x) = -20 \), follow these steps:

### Step 1: Express in terms of tangent
Recall that \( \cot(x) = \frac{1}{\tan(x)} \). Therefore, the equation can be rewritten as:
\[
\frac{1}{\tan(x)} = -20
\]
This implies:
\[
\tan(x) = -\frac{1}{20}
\]

### Step 2: Solve for \( x \)
The general solution for \( \tan(x) = -\frac{1}{20} \) is given by:
\[
x = \arctan\left(-\frac{1}{20}\right) + n\pi, \quad n \in \mathbb{Z}
\]
where \( \arctan\left(-\frac{1}{20}\right) \) is the principal value of the arctangent function.

### Step 3: Calculate the principal value
Using a calculator or trigonometric tables:
\[
\arctan\left(-\frac{1}{20}\right) \approx -0.04996 \text{ radians}
\]

### Step 4: Write the general solution
Therefore, the general solution is:
\[
x \approx -0.04996 + n\pi, \quad n \in \mathbb{Z}
\]

This means that \( x \) can be any of the values obtained by adding multiples of \( \pi \) to \( -0.04996 \).

Would you like further details on this solution or have any additional questions?

Here are 5 related questions you might consider:

1. How does the solution change if \( \cot(x) = 20 \) instead of \( -20 \)?
2. What is the principal value for \( \arctan\left(-\frac{1}{20}\right) \) in degrees?
3. How can you verify the solution by substituting back into the original equation?
4. What are the properties of the cotangent function that affect the solution?
5. How would the solution differ if we had \( \tan(x) = \frac{1}{20} \)?

**Tip:** When solving trigonometric equations, it's crucial to consider the periodicity of the trigonometric functions, which is why the general solution often includes an \( n\pi \) term.

Learn how to solve the trigonometric equation \( \cot(x) = -20 \) with a detailed step-by-step guide. This problem involves trigonometric functions and the arctangent function, providing insights into solving complex equations. Suitable for advanced high school and college students.

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