If `3 sin x + 5 cos x =5`, then what is the value of `(3cos x - 5sin x)` ?

Options

A. 0
B. 2
C. 3
D. 5

Answer

To solve the equation \(3 \sin x + 5 \cos x = 5\) and find the value of \(3 \cos x - 5 \sin x\), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the given equation:
\[ 3 \sin x + 5 \cos x = 5 \]
Squaring both sides gives:
\[ (3 \sin x + 5 \cos x)^2 = 5^2 \]
This simplifies to:
\[ 9 \sin^2 x + 30 \sin x \cos x + 25 \cos^2 x = 25 \]

### Step 2: Rearrange the equation
Now we rearrange the equation:
\[ 9 \sin^2 x + 25 \cos^2 x + 30 \sin x \cos x - 25 = 0 \]

### Step 3: Substitute \(\sin^2 x\) and \(\cos^2 x\)
Using the identity \(\sin^2 x + \cos^2 x = 1\), we can express \(\sin^2 x\) in terms of \(\cos^2 x\):
\[ \sin^2 x = 1 - \cos^2 x \]
Substituting this into the equation gives:
\[ 9(1 - \cos^2 x) + 25 \cos^2 x + 30 \sin x \cos x - 25 = 0 \]
This simplifies to:
\[ 9 - 9 \cos^2 x + 25 \cos^2 x + 30 \sin x \cos x - 25 = 0 \]
Combining like terms results in:
\[ (25 - 9) \cos^2 x + 30 \sin x \cos x - 16 = 0 \]
or
\[ 16 \cos^2 x + 30 \sin x \cos x - 16 = 0 \]

### Step 4: Solve for \(\sin x\) and \(\cos x\)
Now we can express this as a quadratic in terms of \(\sin x\) and \(\cos x\). However, we can also find \(3 \cos x - 5 \sin x\) directly from the original equation.

### Step 5: Find \(3 \cos x - 5 \sin x\)
We can express \(3 \cos x - 5 \sin x\) in terms of \(3 \sin x + 5 \cos x\):
Let \(y = 3 \cos x - 5 \sin x\). We can find the relationship:
\[ y^2 = (3 \cos x - 5 \sin x)^2 = 9 \cos^2 x - 30 \sin x \cos x + 25 \sin^2 x \]
Now, we can add the two squared equations:
\[ (3 \sin x + 5 \cos x)^2 + (3 \cos x - 5 \sin x)^2 = 25 + y^2 \]
Substituting the known values:
\[ 25 + y^2 = 25 + 16 \]
This simplifies to:
\[ y^2 = 16 \]
Thus, we have:
\[ y = \pm 4 \]

### Conclusion
The value of \(3 \cos x - 5 \sin x\) can be either \(4\) or \(-4\).