For the every `theta,0ltthetale90`, find the values of the following angle
`2sin^2theta-3sintheta=-1`

Options

A. `30^@`
B. `45^@`
C. `90^@`
D. both (a) and (c)

Answer

To solve the equation \( 2\sin^2\theta - 3\sin\theta = -1 \) for \( \theta \) in the interval \( 0 < \theta < 90^\circ \), we can follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the equation to set it to zero:
\[ 2\sin^2\theta - 3\sin\theta + 1 = 0 \]

### Step 2: Substitute \( x = \sin\theta \)
Let \( x = \sin\theta \). The equation now becomes:
\[ 2x^2 - 3x + 1 = 0 \]

### Step 3: Factor the quadratic equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to \( 2 \times 1 = 2 \) and add to \( -3 \). The numbers are \( -1 \) and \( -2 \):
\[ 2x^2 - 2x - x + 1 = 0 \]
This can be factored as:
\[ (2x - 1)(x - 1) = 0 \]

### Step 4: Solve for \( x \)
Set each factor to zero:
1. \( 2x - 1 = 0 \) gives \( x = \frac{1}{2} \)
2. \( x - 1 = 0 \) gives \( x = 1 \)

### Step 5: Find \( \theta \)
Now substitute back for \( \sin\theta \):
1. If \( \sin\theta = \frac{1}{2} \), then \( \theta = 30^\circ \).
2. If \( \sin\theta = 1 \), then \( \theta = 90^\circ \).

### Step 6: Conclusion
Thus, the values of \( \theta \) that satisfy the equation \( 2\sin^2\theta - 3\sin\theta = -1 \) in the interval \( 0 < \theta < 90^\circ \) are:
\[ \theta = 30^\circ \quad \text{and} \quad \theta = 90^\circ \]