The arithmetic mean of the numbers `2sin2^(@), 4sin4^(@), 6 sin 6^(@), …………. 178 sin 178^(@), 180 sin 180^(@)`

Options

A. `sin1^(@)`
B. `cot1^(@)`
C. `tan1^(@)`
D. `cos 1^(@)`

Answer

To find the arithmetic mean of the numbers \(2 \sin 2^\circ, 4 \sin 4^\circ, 6 \sin 6^\circ, \ldots, 178 \sin 178^\circ, 180 \sin 180^\circ\), we will follow these steps:

### Step 1: Identify the series
The series consists of terms of the form \(2n \sin(2n^\circ)\) where \(n\) ranges from \(1\) to \(90\). Thus, we can express the series as:
\[ S = 2 \sin 2^\circ + 4 \sin 4^\circ + 6 \sin 6^\circ + \ldots + 178 \sin 178^\circ + 180 \sin 180^\circ \]

### Step 2: Determine the number of terms
The terms range from \(2\) to \(180\) in steps of \(2\). Therefore, the total number of terms is:
\[ n = \frac{180 - 2}{2} + 1 = 90 \]

### Step 3: Simplify the series
Notice that \(\sin 180^\circ = 0\), so we can ignore the last term \(180 \sin 180^\circ\). The remaining terms can be paired:
\[ \sin(180^\circ - k^\circ) = \sin k^\circ \]
This means:
- \(2 \sin 2^\circ\) pairs with \(178 \sin 178^\circ\)
- \(4 \sin 4^\circ\) pairs with \(176 \sin 176^\circ\)
- \(\ldots\)
- \(88 \sin 88^\circ\) pairs with \(92 \sin 92^\circ\)

### Step 4: Pairing the terms
Each pair sums to:
\[ 2n \sin(2n^\circ) + (180 - 2n) \sin(2n^\circ) = 180 \sin(2n^\circ) \]
Thus, we can express the sum \(S\) as:
\[ S = 90 \left( \sin 2^\circ + \sin 4^\circ + \sin 6^\circ + \ldots + \sin 88^\circ \right) + 90 \sin 90^\circ \]
Since \(\sin 90^\circ = 1\), we have:
\[ S = 90 \left( \sin 2^\circ + \sin 4^\circ + \sin 6^\circ + \ldots + \sin 88^\circ + 1 \right) \]

### Step 5: Calculate the sum of the sine terms
The sum of the sine terms can be calculated using the formula for the sum of sines:
\[ \sum_{k=1}^{n} \sin(k \theta) = \frac{\sin\left(\frac{n \theta}{2}\right) \sin\left(\frac{(n+1) \theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \]
In our case, \(n = 44\) (since \(2n\) goes from \(2\) to \(88\)) and \(\theta = 2^\circ\):
\[ \sum_{k=1}^{44} \sin(2k^\circ) = \frac{\sin(44^\circ) \sin(90^\circ)}{\sin(1^\circ)} = \frac{\sin(44^\circ)}{\sin(1^\circ)} \]

### Step 6: Substitute back into the equation for \(S\)
Thus, we can write:
\[ S = 90 \left( \frac{\sin(44^\circ)}{\sin(1^\circ)} + 1 \right) \]

### Step 7: Calculate the arithmetic mean
Finally, the arithmetic mean \(AM\) is given by:
\[ AM = \frac{S}{n} = \frac{90 \left( \frac{\sin(44^\circ)}{\sin(1^\circ)} + 1 \right)}{90} = \frac{\sin(44^\circ)}{\sin(1^\circ)} + 1 \]

### Final Answer
\[ AM = \frac{\sin(44^\circ)}{\sin(1^\circ)} + 1 \]