`"cosec"^(2)theta(cos^(2)theta-3cos theta+2)ge1`, If `theta` belongs to

Options

A. `(0, (pi)/(3))`
B. `((pi)/(2), pi)`
C. `((pi)/(3), (pi)/(2))`
D. `(0, (pi)/(4))`

Answer

To solve the inequality \( \csc^2 \theta (\cos^2 \theta - 3 \cos \theta + 2) \geq 1 \), we will follow these steps:

### Step 1: Rewrite the inequality
We know that \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \). Therefore, we can rewrite the inequality as:
\[ \frac{1}{\sin^2 \theta} (\cos^2 \theta - 3 \cos \theta + 2) \geq 1 \]

### Step 2: Multiply both sides by \( \sin^2 \theta \)
Assuming \( \sin^2 \theta > 0 \) (since \( \theta \) cannot be \( n\pi \), where \( n \) is an integer), we multiply both sides by \( \sin^2 \theta \):
\[ \cos^2 \theta - 3 \cos \theta + 2 \geq \sin^2 \theta \]

### Step 3: Substitute \( \sin^2 \theta \)
Using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \), we can substitute:
\[ \cos^2 \theta - 3 \cos \theta + 2 \geq 1 - \cos^2 \theta \]

### Step 4: Rearrange the inequality
Bringing all terms to one side gives:
\[ \cos^2 \theta + \cos^2 \theta - 3 \cos \theta + 2 - 1 \geq 0 \]
This simplifies to:
\[ 2 \cos^2 \theta - 3 \cos \theta + 1 \geq 0 \]

### Step 5: Factor the quadratic expression
Now we will factor the quadratic:
\[ (2 \cos \theta - 1)(\cos \theta - 1) \geq 0 \]

### Step 6: Find the critical points
Setting each factor to zero gives us the critical points:
1. \( 2 \cos \theta - 1 = 0 \) → \( \cos \theta = \frac{1}{2} \)
2. \( \cos \theta - 1 = 0 \) → \( \cos \theta = 1 \)

### Step 7: Analyze the intervals
The critical points divide the number line into intervals. We will test the sign of the expression in each interval:
1. \( (-\infty, 1) \)
2. \( (1, \frac{1}{2}) \)
3. \( (\frac{1}{2}, \infty) \)

### Step 8: Test values in each interval
- For \( \theta = 0 \) (in \( (-\infty, 1) \)): \( \cos(0) = 1 \) → \( (2(1) - 1)(1 - 1) = 0 \) (satisfies)
- For \( \theta = \frac{\pi}{3} \) (in \( (1, \frac{1}{2}) \)): \( \cos(\frac{\pi}{3}) = \frac{1}{2} \) → \( (2(\frac{1}{2}) - 1)(\frac{1}{2} - 1) = 0 \) (satisfies)
- For \( \theta = \frac{\pi}{4} \) (in \( (\frac{1}{2}, \infty) \)): \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) → \( (2(\frac{\sqrt{2}}{2}) - 1)(\frac{\sqrt{2}}{2} - 1) < 0 \) (does not satisfy)

### Step 9: Determine the solution set
The solution set is where the product is non-negative, which includes the intervals where the expression is zero or positive:
\[ \cos \theta \in (-\infty, 1] \cup [\frac{1}{2}, \infty) \]

### Step 10: Convert back to \( \theta \)
The corresponding values of \( \theta \) are:
- \( \theta = 0 \) to \( \theta = \frac{\pi}{3} \) (for \( \cos \theta = \frac{1}{2} \))
- \( \theta = \frac{\pi}{3} \) to \( \theta = \frac{\pi}{2} \) (for \( \cos \theta = 1 \))

Thus, the final answer is:
\[ \theta \in \left[0, \frac{\pi}{3}\right] \cup \left[\frac{\pi}{2}, \pi\right] \]