Let `f(n, x)=intn cos (nx)dx`, with `f(n, 0)=0.` If the expression `Sigma_(x=1)^(89)f(1, x)` simplifies to `(sina sinb)/(sinc)`, then the value of `(b)/(ac)` is (where `a gt b`)

Options

A. 45
B. 89
C. `(89)/(45)`
D. `(45)/(89)`

Answer

To solve the problem step by step, we start with the function defined as:

\[ f(n, x) = \int n \cos(nx) \, dx \]

### Step 1: Integrate the function
We integrate \( n \cos(nx) \):

\[ f(n, x) = \int n \cos(nx) \, dx = \frac{n}{n} \sin(nx) + C = \sin(nx) + C \]

### Step 2: Apply the condition \( f(n, 0) = 0 \)
We know that \( f(n, 0) = 0 \):

\[ f(n, 0) = \sin(n \cdot 0) + C = 0 \implies C = 0 \]

Thus, we have:

\[ f(n, x) = \sin(nx) \]

### Step 3: Evaluate the summation
We need to evaluate the summation:

\[ \sum_{x=1}^{89} f(1, x) = \sum_{x=1}^{89} \sin(x) \]

### Step 4: Use the formula for the sum of sines
The formula for the sum of sines from \( \sin(1) \) to \( \sin(n) \) is given by:

\[ \sum_{k=1}^{n} \sin(k) = \frac{\sin\left(\frac{n}{2}\right) \sin\left(\frac{n+1}{2}\right)}{\sin\left(\frac{1}{2}\right)} \]

For \( n = 89 \):

\[ \sum_{x=1}^{89} \sin(x) = \frac{\sin\left(\frac{89}{2}\right) \sin\left(\frac{90}{2}\right)}{\sin\left(\frac{1}{2}\right)} \]

### Step 5: Simplify the expression
Calculating the values:

- \( \frac{89}{2} = 44.5 \)
- \( \frac{90}{2} = 45 \)

Thus, we have:

\[ \sum_{x=1}^{89} \sin(x) = \frac{\sin(44.5) \cdot \sin(45)}{\sin(0.5)} \]

### Step 6: Identify the parameters \( a, b, c \)
We compare this with the given expression \( \frac{\sin(a) \sin(b)}{\sin(c)} \):

- \( a = 45 \)
- \( b = 44.5 \)
- \( c = 0.5 \)

### Step 7: Calculate \( \frac{b}{ac} \)
Now, we need to find:

\[ \frac{b}{ac} = \frac{44.5}{45 \cdot 0.5} \]

Calculating the denominator:

\[ 45 \cdot 0.5 = 22.5 \]

Thus, we have:

\[ \frac{b}{ac} = \frac{44.5}{22.5} \]

### Step 8: Simplify the fraction
To simplify \( \frac{44.5}{22.5} \):

\[ \frac{44.5 \times 10}{22.5 \times 10} = \frac{445}{225} \]

### Step 9: Final simplification
Now we can simplify \( \frac{445}{225} \):

\[ \frac{445 \div 5}{225 \div 5} = \frac{89}{45} \]

Thus, the final answer is:

\[ \frac{b}{ac} = \frac{89}{45} \]