The number of solutions of `sin^(2)theta=1/2`in `[0,pi]` is ……

Options

A. three
B. four
C. two
D. one

Explanation

To solve the equation \( \sin^2 \theta = \frac{1}{2} \) in the interval \( [0, \pi] \), we can follow these steps:

### Step 1: Rewrite the equation
Start by taking the square root of both sides of the equation:
\[ \sin \theta = \pm \sqrt{\frac{1}{2}} \]
This simplifies to:
\[ \sin \theta = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \]

**Hint:** Remember that taking the square root introduces both positive and negative solutions.

### Step 2: Analyze the positive solution
In the interval \( [0, \pi] \), the sine function is positive. Therefore, we only consider the positive solution:
\[ \sin \theta = \frac{\sqrt{2}}{2} \]

**Hint:** Identify where the sine function is positive in the given interval.

### Step 3: Find the angles
The sine function equals \( \frac{\sqrt{2}}{2} \) at:
\[ \theta = \frac{\pi}{4} \quad \text{(or 45 degrees)} \]
Since we are looking for solutions in the interval \( [0, \pi] \), we also need to consider the second quadrant where sine is positive.

**Hint:** Recall the sine function's symmetry and the angles in different quadrants.

### Step 4: Find the second angle
The second angle where \( \sin \theta = \frac{\sqrt{2}}{2} \) in the interval \( [0, \pi] \) is:
\[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \quad \text{(or 135 degrees)} \]

**Hint:** Use the property of sine that \( \sin(\pi - \theta) = \sin(\theta) \).

### Step 5: Count the solutions
Thus, the solutions in the interval \( [0, \pi] \) are:
1. \( \theta = \frac{\pi}{4} \)
2. \( \theta = \frac{3\pi}{4} \)

Therefore, the number of solutions is:
\[ \text{Number of solutions} = 2 \]

**Final Answer:** The number of solutions of \( \sin^2 \theta = \frac{1}{2} \) in \( [0, \pi] \) is **2**.