The number of solutions of the equation `sin 2 theta-2 cos theta +4 sin theta=4` in `[0, 5pi]` is equal to

Options

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

Explanation

To solve the equation \( \sin 2\theta - 2\cos\theta + 4\sin\theta = 4 \) in the interval \( [0, 5\pi] \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin 2\theta - 2\cos\theta + 4\sin\theta = 4 \]
Using the double angle identity for sine, we can rewrite \( \sin 2\theta \) as \( 2\sin\theta\cos\theta \):
\[ 2\sin\theta\cos\theta - 2\cos\theta + 4\sin\theta = 4 \]

### Step 2: Rearrange the equation
Next, we can rearrange the equation by moving all terms to one side:
\[ 2\sin\theta\cos\theta + 4\sin\theta - 2\cos\theta - 4 = 0 \]

### Step 3: Factor out common terms
Now, we can factor out \( 2 \):
\[ 2(\sin\theta\cos\theta + 2\sin\theta - \cos\theta - 2) = 0 \]
This simplifies to:
\[ \sin\theta\cos\theta + 2\sin\theta - \cos\theta - 2 = 0 \]

### Step 4: Rearranging further
Rearranging gives:
\[ \sin\theta\cos\theta + 2\sin\theta = \cos\theta + 2 \]

### Step 5: Isolate sine
Now we can isolate \( \sin\theta \):
\[ \sin\theta(\cos\theta + 2) = \cos\theta + 2 \]
If \( \cos\theta + 2 \neq 0 \), we can divide both sides by \( \cos\theta + 2 \):
\[ \sin\theta = 1 \]

### Step 6: Solve for \( \theta \)
The equation \( \sin\theta = 1 \) has solutions:
\[ \theta = \frac{\pi}{2} + 2k\pi \quad (k \in \mathbb{Z}) \]
In the interval \( [0, 5\pi] \), we find the values of \( k \):
- For \( k = 0 \): \( \theta = \frac{\pi}{2} \)
- For \( k = 1 \): \( \theta = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2} \)
- For \( k = 2 \): \( \theta = \frac{\pi}{2} + 4\pi = \frac{9\pi}{2} \)

### Step 7: Check if \( \cos\theta + 2 = 0 \)
Now we check if \( \cos\theta + 2 = 0 \):
\[ \cos\theta = -2 \]
This has no solutions since the range of \( \cos\theta \) is \([-1, 1]\).

### Step 8: Count the solutions
The valid solutions in the interval \( [0, 5\pi] \) are:
1. \( \frac{\pi}{2} \)
2. \( \frac{5\pi}{2} \)
3. \( \frac{9\pi}{2} \)

Thus, the number of solutions is **3**.

### Conclusion
The number of solutions of the equation \( \sin 2\theta - 2\cos\theta + 4\sin\theta = 4 \) in the interval \( [0, 5\pi] \) is **3**.

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