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To solve the problem, let’s examine the given equation:

\[
\sin\frac{\theta}{2} = \sqrt{2} - \sin\frac{\theta}{2}
\]

First, we’ll simplify this equation step-by-step.

### Step 1: Isolate \(\sin\frac{\theta}{2}\)
Bring all terms involving \(\sin\frac{\theta}{2}\) to one side of the equation:

\[
2\sin\frac{\theta}{2} = \sqrt{2}
\]

### Step 2: Solve for \(\sin\frac{\theta}{2}\)
Divide both sides by 2:

\[
\sin\frac{\theta}{2} = \frac{\sqrt{2}}{2}
\]

Since \(\sin\frac{\theta}{2} = \frac{\sqrt{2}}{2}\), we know that:

\[
\frac{\theta}{2} = 45^\circ \quad \text{or} \quad \frac{\theta}{2} = 135^\circ
\]

### Step 3: Find \(\theta\) Values
Multiply both solutions by 2 to find \(\theta\):

1. \(\theta = 2 \times 45^\circ = 90^\circ\)
2. \(\theta = 2 \times 135^\circ = 270^\circ\)

### Step 4: Check the Interval
The interval given is \([0^\circ, 360^\circ)\). Both \(\theta = 90^\circ\) and \(\theta = 270^\circ\) are within this interval.

### Conclusion
Thus, the number of solutions in the interval \([0^\circ, 360^\circ)\) is:

\[
\boxed{2}
\]

Would you like more details on any part of this solution?

Here are some related questions to further explore:

1. How would the solution change if the interval was \([0^\circ, 180^\circ]\) instead?
2. What if the equation involved \(\sin(\theta) = \frac{\sqrt{2}}{2}\) instead of \(\sin\frac{\theta}{2} = \frac{\sqrt{2}}{2}\)?
3. How do we solve similar equations involving \(\cos\) or \(\tan\) functions?
4. What are the properties of the sine function that help in solving such equations?
5. Why is it important to consider the interval \([0^\circ, 360^\circ)\) when solving trigonometric equations?

**Tip:** Always remember to check if your solutions fall within the given interval, as trigonometric functions can produce multiple solutions.

The number of solutions of the equation \( \sin \frac{\theta}{2} = \sqrt{2} - \sin \frac{\theta}{2} \) in the interval \( [0^\circ, 360^\circ) \), is

Discover the step-by-step solution to finding the number of solutions for the trigonometric equation \( \sin \frac{\theta}{2} = \sqrt{2} - \sin \frac{\theta}{2} \) within the interval \( [0^\circ, 360^\circ) \). This problem explores core trigonometric concepts and the process of isolating terms to determine valid solutions.

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