Understanding the Problem
To solve the equation \( \tan(4x) = 0 \), we must recognize that the tangent function equals zero at specific intervals.
Key Points about Tangent Function
- The tangent function is zero at integer multiples of \( \pi \):
- \( \tan(\theta) = 0 \) when \( \theta = n\pi \) where \( n \) is an integer.
Setting Up the Equation
We can express the equation as:
- \( 4x = n\pi \)
Where \( n \) is any integer.
Solving for x
Now, divide both sides by 4 to isolate \( x \):
- \( x = \frac{n\pi}{4} \)
Finding General Solutions
- Since \( n \) can take any integer value, we can express the general solution as:
- \( x = \frac{0\pi}{4}, \frac{1\pi}{4}, \frac{2\pi}{4}, \frac{3\pi}{4}, \ldots \)
Which simplifies to:
- \( x = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \ldots \)
Conclusion
The solutions for the equation \( \tan(4x) = 0 \) can be summarized as:
- General Solution:
- \( x = \frac{n\pi}{4} \) where \( n \in \mathbb{Z} \)
This represents all the angles where the tangent function equals zero, providing a comprehensive understanding of the problem.