The solution set of `(5+4 "cos" theta) (2 "cos" theta +1) =0` in the interval `[0, 2pi]`

Options

A. `{pi//3, 2pi//3}`
B. `{pi//3, pi}`
C. `{2pi//3, 4pi//3}`
D. `{2pi//3, 5pi//3}`

Answer

To solve the equation \((5 + 4 \cos \theta)(2 \cos \theta + 1) = 0\) in the interval \([0, 2\pi]\), we will break it down into two separate equations based on the zero-product property.

### Step 1: Set each factor to zero
We have two factors:
1. \(5 + 4 \cos \theta = 0\)
2. \(2 \cos \theta + 1 = 0\)

### Step 2: Solve the first equation
From the first equation:
\[ 5 + 4 \cos \theta = 0 \]
Subtract 5 from both sides:
\[ 4 \cos \theta = -5 \]
Divide by 4:
\[ \cos \theta = -\frac{5}{4} \]
Since the cosine function has a range of \([-1, 1]\), \(-\frac{5}{4}\) is not a valid solution. Thus, this equation does not contribute any solutions.

### Step 3: Solve the second equation
Now, we solve the second equation:
\[ 2 \cos \theta + 1 = 0 \]
Subtract 1 from both sides:
\[ 2 \cos \theta = -1 \]
Divide by 2:
\[ \cos \theta = -\frac{1}{2} \]

### Step 4: Find the angles for \(\cos \theta = -\frac{1}{2}\)
The cosine function equals \(-\frac{1}{2}\) at specific angles in the interval \([0, 2\pi]\):
\[ \theta = \frac{2\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{3} \]

### Step 5: Write the solution set
Thus, the solution set for the equation \((5 + 4 \cos \theta)(2 \cos \theta + 1) = 0\) in the interval \([0, 2\pi]\) is:
\[ \left\{ \frac{2\pi}{3}, \frac{4\pi}{3} \right\} \]