The number of solutions of the equation` 4sinx – 3cosx = 7 `are

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( 4\sin x - 3\cos x = 7 \) and find the number of solutions, we will follow these steps:

### Step 1: Identify the equation format
The given equation can be expressed in the form \( A\sin x + B\cos x = C \), where:
- \( A = 4 \)
- \( B = -3 \)
- \( C = 7 \)

### Step 2: Calculate \( R \)
We can find \( R \) using the formula:
\[ R = \sqrt{A^2 + B^2} \]
Substituting the values:
\[ R = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 3: Find \( \tan \theta \)
Next, we find \( \tan \theta \) using:
\[ \tan \theta = \frac{A}{B} = \frac{4}{-3} = -\frac{4}{3} \]

### Step 4: Rewrite the equation
We can rewrite the equation \( 4\sin x - 3\cos x = 7 \) in terms of \( R \) and \( \theta \):
\[ R\sin(x + \theta) = C \]
This gives us:
\[ 5\sin(x + \theta) = 7 \]

### Step 5: Solve for \( \sin(x + \theta) \)
Now, we isolate \( \sin(x + \theta) \):
\[ \sin(x + \theta) = \frac{7}{5} \]

### Step 6: Analyze the range of sine
The sine function has a range of \([-1, 1]\). Since \( \frac{7}{5} = 1.4 \) is outside this range, there are no possible values for \( x + \theta \) that satisfy this equation.

### Step 7: Conclusion
Since the equation \( \sin(x + \theta) = \frac{7}{5} \) has no solutions, we conclude that the original equation \( 4\sin x - 3\cos x = 7 \) has no solutions.

Thus, the number of solutions of the equation is **0**.

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