If 3sin x+4 cos x=2, then the value of 3 cos x -4 sin x is equal to:

Options

A. 21
B. `sqrt(21)`
C. `sqrt(23)`
D. `sqrt(29)`

Answer

To solve the problem, we need to find the value of \(3 \cos x - 4 \sin x\) given that \(3 \sin x + 4 \cos x = 2\).

### Step-by-Step Solution:

1. **Given Equation**: Start with the equation provided in the problem:
\[ 3 \sin x + 4 \cos x = 2 \tag{1} \]

2. **Square Both Sides**: We will square both sides of equation (1):
\[ (3 \sin x + 4 \cos x)^2 = 2^2 \]
Expanding the left side:
\[ 9 \sin^2 x + 24 \sin x \cos x + 16 \cos^2 x = 4 \tag{2} \]

3. **Use Pythagorean Identity**: We know that \(\sin^2 x + \cos^2 x = 1\). We can rewrite equation (2) using this identity:
\[ 9 \sin^2 x + 16 \cos^2 x = 9(1 - \cos^2 x) + 16 \cos^2 x \]
Simplifying this gives:
\[ 9 - 9 \cos^2 x + 16 \cos^2 x = 4 \]
Which simplifies to:
\[ 9 + 7 \cos^2 x = 4 \]

4. **Rearranging**: Rearranging the equation gives:
\[ 7 \cos^2 x = 4 - 9 \]
\[ 7 \cos^2 x = -5 \]
This indicates that we made an error in our calculations. Let's go back to equation (2) and simplify correctly.

5. **Correctly Simplifying**: Let's go back to equation (2):
\[ 9 \sin^2 x + 16 \cos^2 x + 24 \sin x \cos x = 4 \]
Using \(\sin^2 x + \cos^2 x = 1\):
\[ 9(1 - \cos^2 x) + 16 \cos^2 x + 24 \sin x \cos x = 4 \]
Simplifying gives:
\[ 9 - 9 \cos^2 x + 16 \cos^2 x + 24 \sin x \cos x = 4 \]
\[ 9 + 7 \cos^2 x + 24 \sin x \cos x = 4 \]
Rearranging gives:
\[ 7 \cos^2 x + 24 \sin x \cos x = 4 - 9 \]
\[ 7 \cos^2 x + 24 \sin x \cos x = -5 \]
This is incorrect.

6. **Finding \(3 \cos x - 4 \sin x\)**: Instead, we can use a different approach. We can express \(3 \cos x - 4 \sin x\) as:
\[ 3 \cos x - 4 \sin x = k \]
Squaring both sides gives:
\[ (3 \cos x - 4 \sin x)^2 = k^2 \]
Expanding gives:
\[ 9 \cos^2 x - 24 \sin x \cos x + 16 \sin^2 x = k^2 \]

7. **Combine Equations**: Now we have two equations:
- From (1): \(9 \sin^2 x + 16 \cos^2 x + 24 \sin x \cos x = 4\)
- From the squared expression: \(9 \cos^2 x + 16 \sin^2 x - 24 \sin x \cos x = k^2\)

8. **Adding the Two Equations**: Adding these equations allows us to eliminate the mixed term:
\[ 9 \sin^2 x + 16 \cos^2 x + 24 \sin x \cos x + 9 \cos^2 x + 16 \sin^2 x - 24 \sin x \cos x = 4 + k^2 \]
Simplifying gives:
\[ 25 = 4 + k^2 \]
Thus:
\[ k^2 = 25 - 4 = 21 \]
Therefore:
\[ k = \sqrt{21} \]

### Final Answer:
The value of \(3 \cos x - 4 \sin x\) is \(\sqrt{21}\).