`2sin^(2)theta-3sintheta=-1`:find`theta`

Options

A. `30^(@)`
B. `45^(@)`
C. `90^(@)`
D. both (1) and (3)

Answer

To solve the equation \( 2\sin^2\theta - 3\sin\theta = -1 \), we will follow these steps:

### Step 1: Rearrange the equation
Start by moving all terms to one side of the equation:
\[ 2\sin^2\theta - 3\sin\theta + 1 = 0 \]

### Step 2: Factor the quadratic equation
We need to factor the quadratic equation \( 2\sin^2\theta - 3\sin\theta + 1 = 0 \). We can look for two numbers that multiply to \( 2 \times 1 = 2 \) and add to \( -3 \). The numbers are \( -2 \) and \( -1 \).

Rewriting the equation:
\[ 2\sin^2\theta - 2\sin\theta - \sin\theta + 1 = 0 \]
Now, we can factor by grouping:
\[ 2\sin\theta(\sin\theta - 1) - 1(\sin\theta - 1) = 0 \]
This gives us:
\[ (2\sin\theta - 1)(\sin\theta - 1) = 0 \]

### Step 3: Set each factor to zero
Now we set each factor equal to zero:
1. \( 2\sin\theta - 1 = 0 \)
2. \( \sin\theta - 1 = 0 \)

### Step 4: Solve each equation
For the first equation:
\[ 2\sin\theta = 1 \implies \sin\theta = \frac{1}{2} \]
The angles that satisfy \( \sin\theta = \frac{1}{2} \) are:
\[ \theta = 30^\circ \quad \text{and} \quad \theta = 150^\circ \]

For the second equation:
\[ \sin\theta = 1 \]
The angle that satisfies \( \sin\theta = 1 \) is:
\[ \theta = 90^\circ \]

### Step 5: Compile all solutions
The solutions for \( \theta \) are:
\[ \theta = 30^\circ, \quad 90^\circ, \quad \text{and} \quad 150^\circ \]

### Final Answer
Thus, the values of \( \theta \) are \( 30^\circ, 90^\circ, \) and \( 150^\circ \).

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