To solve the equation $\cot(x) = -12$, we need to express $\cot(x)$ in terms of trigonometric functions we are more familiar with.

Step 1: Recall the cotangent identity

The cotangent function is the reciprocal of the tangent function, so: $\cot(x) = \frac{1}{\tan(x)}$ Thus, the equation becomes: $\frac{1}{\tan(x)} = -12$ This simplifies to: $\tan(x) = \frac{1}{-12} = -\frac{1}{12}$

Step 2: Find the angle $x$

To find $x$, we take the inverse tangent (or arctangent) of $-\frac{1}{12}$. In other words: $x = \tan^{-1}\left(-\frac{1}{12}\right)$ Using a calculator: $x \approx -4.76^\circ$

Step 3: Find all possible solutions

Since the tangent function is periodic with a period of $\pi$ radians (or 180°), the general solution for $x$ will be: x = -4.76^\circ + 180^\circ n \quad \text{(for any integer n)} or in radians: x = -0.083 \, \text{radians} + \pi n \quad \text{(for any integer n)}

Final Answer:

The values of $x$ that satisfy $\cot(x) = -12$ are approximately: x \approx -4.76^\circ + 180^\circ n \quad \text{(for any integer n)} or x \approx -0.083 \, \text{radians} + \pi n \quad \text{(for any integer n)}

Would you like to see a graphical representation or any more details on solving trigonometric equations?

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Here are 5 related questions you could explore next:

1. What is the solution for $\tan(x) = 12$?
2. How do we solve trigonometric equations with multiple functions involved?
3. What is the difference between solving for $\cot(x)$ and $\tan(x)$?
4. How does the periodicity of trigonometric functions affect their solutions?
5. What are some common applications of trigonometric equations in real-world problems?

Tip: When solving trigonometric equations, always consider the periodic nature of trigonometric functions, as they often have infinite solutions based on their periodicity.