The number of values of `theta` satisfying `4 cos theta+3 sin theta=5` as well as `3 cos theta + 4 sin theta = 5` is

Options

A. one
B. two
C. zero
D. none of these

Explanation

To solve the problem, we need to find the number of values of \( \theta \) that satisfy the two equations:

1. \( 4 \cos \theta + 3 \sin \theta = 5 \) (Equation 1)
2. \( 3 \cos \theta + 4 \sin \theta = 5 \) (Equation 2)

### Step-by-Step Solution

**Step 1: Analyze the first equation.**

From Equation 1:
\[ 4 \cos \theta + 3 \sin \theta = 5 \]

**Step 2: Analyze the second equation.**

From Equation 2:
\[ 3 \cos \theta + 4 \sin \theta = 5 \]

**Step 3: Add both equations.**

Adding Equation 1 and Equation 2:
\[ (4 \cos \theta + 3 \sin \theta) + (3 \cos \theta + 4 \sin \theta) = 5 + 5 \]
This simplifies to:
\[ (4 + 3) \cos \theta + (3 + 4) \sin \theta = 10 \]
\[ 7 \cos \theta + 7 \sin \theta = 10 \]

**Step 4: Factor out the common term.**

Factoring out 7:
\[ 7 (\cos \theta + \sin \theta) = 10 \]

**Step 5: Isolate \( \cos \theta + \sin \theta \).**

Dividing both sides by 7:
\[ \cos \theta + \sin \theta = \frac{10}{7} \]

**Step 6: Determine the maximum value of \( \cos \theta + \sin \theta \).**

The maximum value of \( \cos \theta + \sin \theta \) can be found using the identity:
\[ \cos \theta + \sin \theta \leq \sqrt{1^2 + 1^2} = \sqrt{2} \]

**Step 7: Compare the values.**

Now we check if \( \frac{10}{7} \) is less than or equal to \( \sqrt{2} \):
\[ \sqrt{2} \approx 1.414 \]
Calculating \( \frac{10}{7} \):
\[ \frac{10}{7} \approx 1.42857 \]

Since \( \frac{10}{7} > \sqrt{2} \), it is impossible for \( \cos \theta + \sin \theta \) to equal \( \frac{10}{7} \).

**Step 8: Conclusion.**

Thus, there are no values of \( \theta \) that satisfy both equations simultaneously.

### Final Answer
The number of values of \( \theta \) satisfying both equations is **0**.