Solve the trigonometric equation for all values minus, 4, is less than, x, is less than, 4−4<x<4.
sine, one quarter, x, equals, 0
sin 
4
1
​	
 x=0

The trigonometric equation you provided is:

\[
\sin\left(\frac{1}{4}x\right) = 0
\]

We need to find all solutions for \( x \) within the interval \(-4 < x < 4\).

### Step 1: Solve the basic equation
The sine function equals zero at the following angles:

\[
\frac{1}{4}x = n\pi \quad \text{where } n \text{ is an integer}
\]

### Step 2: Solve for \( x \)
To find \( x \), multiply both sides by 4:

\[
x = 4n\pi
\]

### Step 3: Determine \( n \) such that \(-4 < x < 4\)
Substitute \( x = 4n\pi \) into the inequality \(-4 < x < 4\):

\[
-4 < 4n\pi < 4
\]

Divide by 4:

\[
-\frac{1}{\pi} < n < \frac{1}{\pi}
\]

Since \(\frac{1}{\pi} \approx 0.3183\), \( n \) can only be 0 (as \( n \) must be an integer).

### Step 4: Verify the solution
For \( n = 0 \):

\[
x = 4(0)\pi = 0
\]

So, \( x = 0 \) is the only solution within the interval \(-4 < x < 4\).

### Final Answer:
The solution to the equation \(\sin\left(\frac{1}{4}x\right) = 0\) within the interval \(-4 < x < 4\) is:

\[
x = 0
\]

Would you like more details, or do you have any other questions?

### Related Questions:
1. How would the solution change if the interval was different?
2. What are the general solutions to \(\sin(kx) = 0\) for any constant \( k \)?
3. How can we solve similar equations involving \(\cos(x)\) or \(\tan(x)\)?
4. What is the significance of the principal value in trigonometric equations?
5. How do the solutions change if the equation involved \(\sin(x) = 1\) instead of \(0\)?

### Tip:
When solving trigonometric equations, always check if the interval provided limits the number of solutions.

Explore how to solve the trigonometric equation sine (1/4)x = 0 within the interval -4 < x < 4. This problem delves into trigonometric functions and their properties, suitable for students in Grades 10-12.

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