Find the values of `theta` between `0^(@)` and `360^(@)` which satisfy the equations
`sin^(2)theta=(3)/(4)`

Answer

To solve the equation \( \sin^2 \theta = \frac{3}{4} \) for \( \theta \) in the range \( 0^\circ \) to \( 360^\circ \), we can follow these steps:

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

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

### Step 3: Identify the cases
Now we have two cases to consider:
1. \( \sin \theta = \frac{\sqrt{3}}{2} \)
2. \( \sin \theta = -\frac{\sqrt{3}}{2} \)

### Step 4: Solve the first case
For the first case \( \sin \theta = \frac{\sqrt{3}}{2} \):
- The angles where \( \sin \theta = \frac{\sqrt{3}}{2} \) in the range \( 0^\circ \) to \( 360^\circ \) are:
\[ \theta = 60^\circ \quad \text{(1st quadrant)} \]
\[ \theta = 120^\circ \quad \text{(2nd quadrant)} \]

### Step 5: Solve the second case
For the second case \( \sin \theta = -\frac{\sqrt{3}}{2} \):
- The angles where \( \sin \theta = -\frac{\sqrt{3}}{2} \) in the range \( 0^\circ \) to \( 360^\circ \) are:
\[ \theta = 240^\circ \quad \text{(3rd quadrant)} \]
\[ \theta = 300^\circ \quad \text{(4th quadrant)} \]

### Step 6: Compile the solutions
Combining the solutions from both cases, we have:
\[ \theta = 60^\circ, 120^\circ, 240^\circ, 300^\circ \]

### Final Answer
Thus, the values of \( \theta \) that satisfy the equation \( \sin^2 \theta = \frac{3}{4} \) in the range \( 0^\circ \) to \( 360^\circ \) are:
\[ \theta = 60^\circ, 120^\circ, 240^\circ, 300^\circ \]

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