If the equation `(cos theta - 1) x^(2) + (cos theta ) x + sin theta =0` has real roots, then `theta` lies in

Options

A. `[0, pi]`
B. `[-pi, 0]`
C. `[-(pi)/(2),(pi)/(2)]`
D. `[0, 2 pi]`

Answer

To determine the values of \(\theta\) for which the equation

\[ (\cos \theta - 1)x^2 + (\cos \theta)x + \sin \theta = 0 \]

has real roots, we need to analyze the discriminant of the quadratic equation. The discriminant \(D\) for a quadratic equation \(ax^2 + bx + c = 0\) is given by:

\[ D = b^2 - 4ac \]

### Step 1: Identify coefficients
In our equation, we can identify:
- \(a = \cos \theta - 1\)
- \(b = \cos \theta\)
- \(c = \sin \theta\)

### Step 2: Write the discriminant
Now, we can write the discriminant:

\[ D = (\cos \theta)^2 - 4(\cos \theta - 1)(\sin \theta) \]

### Step 3: Expand the discriminant
Expanding the expression for the discriminant:

\[ D = \cos^2 \theta - 4(\cos \theta \sin \theta - \sin \theta) \]
\[ = \cos^2 \theta - 4\cos \theta \sin \theta + 4\sin \theta \]

### Step 4: Set the discriminant greater than or equal to zero
For the quadratic equation to have real roots, the discriminant must be greater than or equal to zero:

\[ \cos^2 \theta - 4\cos \theta \sin \theta + 4\sin \theta \geq 0 \]

### Step 5: Analyze the inequality
This is a quadratic inequality in terms of \(\cos \theta\). We can treat it as:

\[ a = 1, \quad b = -4\sin \theta, \quad c = 4\sin \theta \]

The roots of this quadratic can be found using the quadratic formula:

\[ \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 6: Calculate the discriminant of the quadratic in \(\cos \theta\)
The discriminant of this quadratic must also be non-negative:

\[ (-4\sin \theta)^2 - 4(1)(4\sin \theta) \geq 0 \]
\[ 16\sin^2 \theta - 16\sin \theta \geq 0 \]
\[ 16\sin \theta(\sin \theta - 1) \geq 0 \]

### Step 7: Solve the inequality
The critical points are \(\sin \theta = 0\) and \(\sin \theta = 1\). The intervals to test are:
1. \((-\infty, 0)\)
2. \((0, 1)\)
3. \((1, \infty)\)

### Step 8: Determine valid intervals
- For \(0 \leq \sin \theta \leq 1\), the product \(16\sin \theta(\sin \theta - 1) \geq 0\) is satisfied.
- Therefore, \(\theta\) must lie in the intervals where \(\sin \theta\) is non-negative, which corresponds to:

\[ \theta \in [0, \frac{\pi}{2}] \cup [\frac{3\pi}{2}, 2\pi] \]

### Conclusion
Thus, the values of \(\theta\) for which the equation has real roots are:

\[ \theta \in [0, \frac{\pi}{2}] \cup [\frac{3\pi}{2}, 2\pi] \]