The value of `[100(x-1)]` is where [x] is the greatest integer less than or equal to x and `x=(sum_(n=1)^44 cos n^@)/(sum_(n=1)^44 sin n^@)`

Options

A. 140
B. 141
C. 142
D. 144

Answer

To solve the problem step by step, we need to find the value of \( x \) given by the formula:

\[ x = \frac{\sum_{n=1}^{44} \cos n^\circ}{\sum_{n=1}^{44} \sin n^\circ} \]

### Step 1: Calculate the sums of cosines and sines

We start by calculating the sums in the numerator and denominator separately.

**Numerator:**
\[ \sum_{n=1}^{44} \cos n^\circ \]

**Denominator:**
\[ \sum_{n=1}^{44} \sin n^\circ \]

### Step 2: Use the formula for the sum of sines and cosines

Using the formulas for the sums of sines and cosines, we have:

\[ \sum_{n=1}^{N} \cos n^\circ = \frac{\sin\left(\frac{N}{2} \cdot 1^\circ\right) \cdot \cos\left(\frac{(N+1) \cdot 1^\circ}{2}\right)}{\sin\left(\frac{1^\circ}{2}\right)} \]

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

For \( N = 44 \):

**Numerator:**
\[ \sum_{n=1}^{44} \cos n^\circ = \frac{\sin\left(22^\circ\right) \cdot \cos\left(22.5^\circ\right)}{\sin\left(0.5^\circ\right)} \]

**Denominator:**
\[ \sum_{n=1}^{44} \sin n^\circ = \frac{\sin\left(22^\circ\right) \cdot \sin\left(22.5^\circ\right)}{\sin\left(0.5^\circ\right)} \]

### Step 3: Substitute the sums into the equation for \( x \)

Now substituting these into the expression for \( x \):

\[ x = \frac{\frac{\sin(22^\circ) \cdot \cos(22.5^\circ)}{\sin(0.5^\circ)}}{\frac{\sin(22^\circ) \cdot \sin(22.5^\circ)}{\sin(0.5^\circ)}} \]

This simplifies to:

\[ x = \frac{\cos(22.5^\circ)}{\sin(22.5^\circ)} = \cot(22.5^\circ) \]

### Step 4: Calculate \( \cot(22.5^\circ) \)

Using the half-angle formula:

\[ \cot(22.5^\circ) = \frac{1 + \cos(45^\circ)}{\sin(45^\circ)} = \frac{1 + \frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = \sqrt{2} + 1 \]

### Step 5: Substitute \( x \) into the expression for \( 100(x - 1) \)

Now we need to calculate:

\[ 100(x - 1) = 100\left((\sqrt{2} + 1) - 1\right) = 100\sqrt{2} \]

### Step 6: Find the greatest integer less than or equal to \( 100\sqrt{2} \)

Using the approximate value of \( \sqrt{2} \approx 1.414 \):

\[ 100\sqrt{2} \approx 100 \times 1.414 = 141.4 \]

Thus, the greatest integer less than or equal to \( 141.4 \) is \( 141 \).

### Final Answer

\[ \lfloor 100(x - 1) \rfloor = 141 \]