Ctg x > − 1
To solve the inequality \( \cot x > -1 \), we need to analyze the behavior of the cotangent function.
Understanding the cotangent function: The cotangent function \( \cot x = \frac{\cos x}{\sin x} \) is defined wherever \( \sin x \neq 0 \). This means cotangent is undefined at integer multiples of \( \pi \) (i.e., \( x = n\pi \) for \( n \in \mathbb{Z} \)).
Finding the region of interest: We want to determine where \( \cot x > -1 \). We can rewrite the inequality: \[ \cot x + 1 > 0 \implies \cot x > -1 \]
Analyzing the condition: This inequality can be solved by considering when: \[ \frac{\cos x}{\sin x} > -1 \] Rearranging gives: \[ \cos x > -\sin x \] or equivalently, \[ \cos x + \sin x > 0 \]
Finding the angle: The expression \( \cos x + \sin x = 0 \) has solutions: \[ x = \frac{3\pi}{4} + n\pi, \quad n \in \mathbb{Z} \] This is where the line \( y = \cos x \) crosses \( y = -\sin x \).
Testing intervals: Next, we check the intervals split by those critical points:
- For \( x < \frac{3\pi}{4} \), \( \cos x + \sin x > 0 \)
- For \( x = \frac{3\pi}{4} \), \( \cos x + \sin x = 0 \)
- For \( x > \frac{3\pi}{4} \) and \( x < \frac{7\pi}{4} \), \( \cos x + \sin x < 0 \)
Conclusions: Thus, the solution set can be expressed: \[ x \in \left( n\pi, \frac{3\pi}{4} + n\pi \right) \text{ and } x \in \left( \frac{7\pi}{4} + n\pi, (n+1)\pi \right) \] for all integers \( n \).
The final answer is: \[ x \in \bigcup_{n \in \mathbb{Z}} \left( n\pi, \frac{3\pi}{4} + n\pi \right) \cup \left( \frac{7\pi}{4} + n\pi, (n+1)\pi \right) \]
