`sin^2 theta - 3 sin theta +2=0` will be true if

Options

A. `0 ge theta gt 90`
B. `0 gt theta gt 90`
C. `theta =0^@`
D. `theta =90^@`

Answer

To solve the equation \( \sin^2 \theta - 3 \sin \theta + 2 = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
The equation is already in a standard quadratic form:
\[ \sin^2 \theta - 3 \sin \theta + 2 = 0 \]

### Step 2: Factor the quadratic equation
We need to factor the quadratic expression. We look for two numbers that multiply to \( 2 \) (the constant term) and add up to \( -3 \) (the coefficient of \( \sin \theta \)). The numbers that satisfy this are \( -1 \) and \( -2 \). Therefore, we can factor the equation as:
\[ (\sin \theta - 1)(\sin \theta - 2) = 0 \]

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

### Step 4: Solve for \( \sin \theta \)
From the first equation:
\[ \sin \theta = 1 \]
From the second equation:
\[ \sin \theta = 2 \]

### Step 5: Determine valid solutions
The sine function has a range of \([-1, 1]\). Therefore, \( \sin \theta = 2 \) is not a valid solution since it is outside the range of the sine function.

The only valid solution is:
\[ \sin \theta = 1 \]

### Step 6: Find the angle \( \theta \)
The angle \( \theta \) for which \( \sin \theta = 1 \) is:
\[ \theta = 90^\circ \]

### Conclusion
Thus, the solution to the equation \( \sin^2 \theta - 3 \sin \theta + 2 = 0 \) is:
\[ \theta = 90^\circ \]