The number of solutions of equations `3cos2theta+5costheta =1` in `[0,2pi]` is

Options

A. 8
B. 6
C. 4
D. 2

Answer

To solve the equation \(3 \cos 2\theta + 5 \cos \theta = 1\) in the interval \([0, 2\pi]\), we will follow these steps:

### Step 1: Use the double angle formula
We know that \(\cos 2\theta = 2 \cos^2 \theta - 1\). We can substitute this into the equation:

\[ 3(2 \cos^2 \theta - 1) + 5 \cos \theta = 1 \]

### Step 2: Simplify the equation
Expanding the equation gives:

\[ 6 \cos^2 \theta - 3 + 5 \cos \theta = 1 \]

Now, rearranging the equation:

\[ 6 \cos^2 \theta + 5 \cos \theta - 4 = 0 \]

### Step 3: Identify the quadratic form
This is a quadratic equation in terms of \(\cos \theta\):

\[ 6x^2 + 5x - 4 = 0 \]

where \(x = \cos \theta\).

### Step 4: Apply the quadratic formula
The quadratic formula is given by:

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

Here, \(a = 6\), \(b = 5\), and \(c = -4\). Plugging in these values:

\[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 6 \cdot (-4)}}{2 \cdot 6} \]

Calculating the discriminant:

\[ b^2 - 4ac = 25 + 96 = 121 \]

Now substituting back:

\[ x = \frac{-5 \pm \sqrt{121}}{12} \]

### Step 5: Calculate the roots
Since \(\sqrt{121} = 11\):

\[ x = \frac{-5 \pm 11}{12} \]

Calculating the two possible values:

1. \(x_1 = \frac{-5 + 11}{12} = \frac{6}{12} = \frac{1}{2}\)
2. \(x_2 = \frac{-5 - 11}{12} = \frac{-16}{12} = -\frac{4}{3}\)

### Step 6: Check the validity of the roots
The range of \(\cos \theta\) is \([-1, 1]\).

- The root \(\frac{1}{2}\) is valid.
- The root \(-\frac{4}{3}\) is invalid since it is less than -1.

### Step 7: Find the angles corresponding to valid roots
Now we need to find the angles \(\theta\) for \(\cos \theta = \frac{1}{2}\):

\[ \theta = \frac{\pi}{3} \quad \text{and} \quad \theta = \frac{5\pi}{3} \]

### Step 8: Count the solutions
The solutions in the interval \([0, 2\pi]\) are:

1. \(\theta = \frac{\pi}{3}\)
2. \(\theta = \frac{5\pi}{3}\)

Thus, the total number of solutions is **2**.

### Final Answer
The number of solutions of the equation \(3 \cos 2\theta + 5 \cos \theta = 1\) in the interval \([0, 2\pi]\) is **2**.

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