The number of values of `theta` between `-pi and (3pi)/(2)` that satisfies the equation `5cos 2 theta+2 "cos"^(2)(theta)/(2)+1=0` is :

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the equation \(5\cos(2\theta) + 2\cos^2(\theta) + 1 = 0\) for the number of values of \(\theta\) between \(-\pi\) and \(\frac{3\pi}{2}\), we can follow these steps:

### Step 1: Use the double angle identity for cosine
The double angle identity states that:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
Substituting this into the equation gives:
\[ 5(2\cos^2(\theta) - 1) + 2\cos^2(\theta) + 1 = 0 \]

### Step 2: Simplify the equation
Expanding the equation:
\[ 10\cos^2(\theta) - 5 + 2\cos^2(\theta) + 1 = 0 \]
Combining like terms:
\[ (10\cos^2(\theta) + 2\cos^2(\theta)) - 5 + 1 = 0 \]
This simplifies to:
\[ 12\cos^2(\theta) - 4 = 0 \]

### Step 3: Rearranging the equation
Rearranging gives:
\[ 12\cos^2(\theta) = 4 \]
Dividing both sides by 12:
\[ \cos^2(\theta) = \frac{1}{3} \]

### Step 4: Solving for \(\cos(\theta)\)
Taking the square root of both sides:
\[ \cos(\theta) = \pm \frac{1}{\sqrt{3}} = \pm \frac{\sqrt{3}}{3} \]

### Step 5: Finding the angles
Now we need to find the angles \(\theta\) that satisfy \(\cos(\theta) = \frac{\sqrt{3}}{3}\) and \(\cos(\theta) = -\frac{\sqrt{3}}{3}\).

1. For \(\cos(\theta) = \frac{\sqrt{3}}{3}\):
- The angles are:
\[ \theta = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad \theta = -\frac{\pi}{6} + 2k\pi \]

2. For \(\cos(\theta) = -\frac{\sqrt{3}}{3}\):
- The angles are:
\[ \theta = \frac{5\pi}{6} + 2k\pi \quad \text{and} \quad \theta = -\frac{5\pi}{6} + 2k\pi \]

### Step 6: Finding valid solutions in the interval \([- \pi, \frac{3\pi}{2}]\)
Now we need to find all valid solutions for \(k = 0\) and \(k = -1\):

- For \(k = 0\):
- \(\theta = \frac{\pi}{6}\)
- \(\theta = -\frac{\pi}{6}\)
- \(\theta = \frac{5\pi}{6}\)
- \(\theta = -\frac{5\pi}{6}\)

- For \(k = -1\):
- \(\theta = \frac{\pi}{6} - 2\pi = -\frac{11\pi}{6}\) (not in the interval)
- \(\theta = -\frac{\pi}{6} - 2\pi = -\frac{13\pi}{6}\) (not in the interval)
- \(\theta = \frac{5\pi}{6} - 2\pi = -\frac{7\pi}{6}\) (not in the interval)
- \(\theta = -\frac{5\pi}{6} - 2\pi = -\frac{17\pi}{6}\) (not in the interval)

### Step 7: Count the valid solutions
The valid solutions in the interval \([- \pi, \frac{3\pi}{2}]\) are:
1. \(-\frac{5\pi}{6}\)
2. \(-\frac{\pi}{6}\)
3. \(\frac{\pi}{6}\)
4. \(\frac{5\pi}{6}\)

Thus, there are a total of **4 solutions**.

### Final Answer
The number of values of \(\theta\) between \(-\pi\) and \(\frac{3\pi}{2}\) that satisfy the equation is **4**.