Find the number of integral values of k for which the equation `7 cos x+5 sin x=2k+1` has at least one solution.

Explanation

To find the number of integral values of \( k \) for which the equation \( 7 \cos x + 5 \sin x = 2k + 1 \) has at least one solution, we will follow these steps:

### Step 1: Determine the range of \( 7 \cos x + 5 \sin x \)

The expression \( 7 \cos x + 5 \sin x \) can be rewritten in terms of its maximum and minimum values. The maximum value of \( a \cos x + b \sin x \) is given by \( \sqrt{a^2 + b^2} \).

Here, \( a = 7 \) and \( b = 5 \).

\[ \text{Maximum value} = \sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74} \]

The minimum value will be \( -\sqrt{74} \).

### Step 2: Set up the inequalities

Since \( 7 \cos x + 5 \sin x \) can take values between \( -\sqrt{74} \) and \( \sqrt{74} \), we can set up the following inequalities for \( 2k + 1 \):

\[ -\sqrt{74} \leq 2k + 1 \leq \sqrt{74} \]

### Step 3: Solve the inequalities for \( k \)

First, we will solve the left inequality:

\[ -\sqrt{74} \leq 2k + 1 \]

Subtracting 1 from both sides:

\[ -\sqrt{74} - 1 \leq 2k \]

Dividing by 2:

\[ \frac{-\sqrt{74} - 1}{2} \leq k \]

Now, solving the right inequality:

\[ 2k + 1 \leq \sqrt{74} \]

Subtracting 1 from both sides:

\[ 2k \leq \sqrt{74} - 1 \]

Dividing by 2:

\[ k \leq \frac{\sqrt{74} - 1}{2} \]

### Step 4: Calculate the numerical values

Now we need to calculate the numerical values of \( \sqrt{74} \):

\[ \sqrt{74} \approx 8.6 \]

Thus, we have:

\[ \frac{-8.6 - 1}{2} \leq k \leq \frac{8.6 - 1}{2} \]

Calculating these:

\[ \frac{-9.6}{2} \leq k \leq \frac{7.6}{2} \]

This simplifies to:

\[ -4.8 \leq k \leq 3.8 \]

### Step 5: Determine the integral values of \( k \)

The integral values of \( k \) that satisfy this inequality are:

\[ k = -4, -3, -2, -1, 0, 1, 2, 3 \]

### Step 6: Count the integral values

Counting these values gives us a total of 8 integral values.

### Final Answer

The number of integral values of \( k \) for which the equation \( 7 \cos x + 5 \sin x = 2k + 1 \) has at least one solution is **8**.

---