We are given the equation:

\[
-4 + 4 \sin\left(\frac{\theta}{4}\right) = 0
\]

Let's solve for \(\theta\):

### Step 1: Isolate the sine term
Add 4 to both sides to eliminate the constant term on the left:

\[
4 \sin\left(\frac{\theta}{4}\right) = 4
\]

### Step 2: Simplify the equation
Divide both sides by 4:

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

### Step 3: Solve for \(\frac{\theta}{4}\)
We know that \(\sin x = 1\) when \(x = \frac{\pi}{2} + 2n\pi\), where \(n\) is an integer. So:

\[
\frac{\theta}{4} = \frac{\pi}{2} + 2n\pi
\]

### Step 4: Solve for \(\theta\)
Multiply both sides by 4:

\[
\theta = 2\pi + 8n\pi
\]

Thus, the general solution is:

\[
\theta = 2\pi(1 + 4n), \quad n \in \mathbb{Z}
\]

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How does the sine function behave for different angles?
2. What are the general solutions for trigonometric equations involving sine?
3. How would the solution change if the equation was \(-4 + 4 \sin(\theta/4) = 2\)?
4. What is the period of the sine function and how does it affect solutions?
5. How would you solve for \(\theta\) if the sine function had a different argument, like \(\sin(\theta/6)\)?

**Tip**: When solving trigonometric equations, always consider the general form of the solution, which includes the periodic nature of the sine and cosine functions.

Learn how to solve the trigonometric equation -4 + 4sin(theta/4) = 0 with a detailed, step-by-step solution. This problem involves key trigonometric concepts, including solving for theta and understanding the periodic nature of sine functions. Suitable for students in Grades 10-12.

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