Find the general value of ` theta` which satisfies the equation `sin ^(2) theta = (3)/(4).`

Answer

To find the general value of \( \theta \) that satisfies the equation \( \sin^2 \theta = \frac{3}{4} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin^2 \theta = \frac{3}{4} \]

### Step 2: Take the square root
Taking the square root of both sides, we get:
\[ \sin \theta = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \]

### Step 3: Find the angles corresponding to the sine values
The sine function equals \( \frac{\sqrt{3}}{2} \) at:
\[ \theta = \frac{\pi}{3} + 2n\pi \quad \text{(for the positive value)} \]
And it equals \( -\frac{\sqrt{3}}{2} \) at:
\[ \theta = \frac{4\pi}{3} + 2n\pi \quad \text{(for the negative value)} \]

### Step 4: Combine the solutions
Thus, the general solutions for \( \theta \) can be combined as:
\[ \theta = \frac{\pi}{3} + 2n\pi \quad \text{and} \quad \theta = \frac{4\pi}{3} + 2n\pi \]

### Step 5: Write the final answer
Therefore, the general values of \( \theta \) that satisfy the equation \( \sin^2 \theta = \frac{3}{4} \) are:
\[ \theta = \frac{\pi}{3} + 2n\pi \quad \text{and} \quad \theta = \frac{4\pi}{3} + 2n\pi \quad (n \in \mathbb{Z}) \]

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