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 need to evaluate the function \( f(n, x) \) and then compute the sum \( \Sigma_{x=1}^{89} f(1, x) \).

### Step 1: Evaluate the function \( f(n, x) \)

The function is defined as:
\[ f(n, x) = \int \cos(nx) \, dx \]

To integrate \( \cos(nx) \), we have:
\[ \int \cos(nx) \, dx = \frac{1}{n} \sin(nx) + C \]

### Step 2: Determine the constant \( C \)

We are given that \( f(n, 0) = 0 \). Substituting \( x = 0 \) into the expression we found:
\[ f(n, 0) = \frac{1}{n} \sin(n \cdot 0) + C = \frac{1}{n} \cdot 0 + C = C \]
Since \( f(n, 0) = 0 \), we have \( C = 0 \). Therefore:
\[ f(n, x) = \frac{1}{n} \sin(nx) \]

### Step 3: Compute \( f(1, x) \)

Substituting \( n = 1 \):
\[ f(1, x) = \sin(x) \]

### Step 4: Compute the sum \( \Sigma_{x=1}^{89} f(1, x) \)

Now we need to evaluate:
\[ \Sigma_{x=1}^{89} f(1, x) = \Sigma_{x=1}^{89} \sin(x) \]

### Step 5: Use the formula for the sum of sines

The sum of sines can be computed using the formula:
\[ \Sigma_{k=1}^{n} \sin(k) = \frac{\sin\left(\frac{n}{2}\right) \sin\left(\frac{n+1}{2}\right)}{\sin\left(\frac{1}{2}\right)} \]
For \( n = 89 \):
\[ \Sigma_{x=1}^{89} \sin(x) = \frac{\sin\left(\frac{89}{2}\right) \sin\left(\frac{90}{2}\right)}{\sin\left(\frac{1}{2}\right)} \]
This simplifies to:
\[ \Sigma_{x=1}^{89} \sin(x) = \frac{\sin(44.5) \cdot 1}{\sin(0.5)} \]

### Step 6: Compare with the given expression

We are told that this expression simplifies to:
\[ \frac{\sin a \sin b}{\sin c} \]
From our result:
\[ \frac{\sin(44.5) \cdot 1}{\sin(0.5)} \]
We can identify:
- \( a = 44.5 \)
- \( b = 1 \)
- \( c = 0.5 \)

### Step 7: Find the value of \( \frac{b}{ac} \)

Now we compute \( \frac{b}{ac} \):
\[ \frac{b}{ac} = \frac{1}{44.5 \cdot 0.5} = \frac{1}{22.25} \]

### Final Calculation

To express \( \frac{1}{22.25} \) in terms of fractions:
\[ \frac{1}{22.25} = \frac{4}{89} \]

### Conclusion

Thus, the value of \( \frac{b}{ac} \) is:
\[ \frac{4}{89} \]