The value of the expression `sin^(2)1^(@)+sin^(2)11^(@)+sin^(2)21^(@)+`
`sin^(2)31^(@)+sin^(2)41^(@)+sin^(2)45^(@)+`
`sin^(2)49^(@)+sin^(2)59^(@)+sin^(2)69^(@)+`
`sin^(2)79^(@)+sin^(2)89^(@)` is :

Options

A. 0
B. `5(1)/(2)`
C. 5
D. `4(1)/(2)`

Answer

To solve the expression

\[ \sin^2(1^\circ) + \sin^2(11^\circ) + \sin^2(21^\circ) + \sin^2(31^\circ) + \sin^2(41^\circ) + \sin^2(45^\circ) + \sin^2(51^\circ) + \sin^2(61^\circ) + \sin^2(71^\circ) + \sin^2(79^\circ) + \sin^2(89^\circ) \]

we can use the identity that relates sine and cosine:

\[ \sin^2(\theta) = \cos^2(90^\circ - \theta) \]

### Step 1: Rewrite the expression using the identity

We can pair terms in the expression using the identity. For example:

- \(\sin^2(1^\circ) = \cos^2(89^\circ)\)
- \(\sin^2(11^\circ) = \cos^2(79^\circ)\)
- \(\sin^2(21^\circ) = \cos^2(69^\circ)\)
- \(\sin^2(31^\circ) = \cos^2(59^\circ)\)
- \(\sin^2(41^\circ) = \cos^2(49^\circ)\)
- \(\sin^2(45^\circ) = \frac{1}{2}\)

Thus, we can rewrite the expression as:

\[ \sin^2(1^\circ) + \sin^2(11^\circ) + \sin^2(21^\circ) + \sin^2(31^\circ) + \sin^2(41^\circ) + \frac{1}{2} + \sin^2(51^\circ) + \sin^2(61^\circ) + \sin^2(71^\circ) + \sin^2(79^\circ) + \sin^2(89^\circ) \]

### Step 2: Group the terms

Now, we can group the terms:

\[ (\sin^2(1^\circ) + \sin^2(89^\circ)) + (\sin^2(11^\circ) + \sin^2(79^\circ)) + (\sin^2(21^\circ) + \sin^2(69^\circ)) + (\sin^2(31^\circ) + \sin^2(59^\circ)) + (\sin^2(41^\circ) + \sin^2(49^\circ)) + \sin^2(45^\circ) \]

### Step 3: Apply the identity

Using the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\), we can simplify each pair:

- \(\sin^2(1^\circ) + \sin^2(89^\circ) = 1\)
- \(\sin^2(11^\circ) + \sin^2(79^\circ) = 1\)
- \(\sin^2(21^\circ) + \sin^2(69^\circ) = 1\)
- \(\sin^2(31^\circ) + \sin^2(59^\circ) = 1\)
- \(\sin^2(41^\circ) + \sin^2(49^\circ) = 1\)

### Step 4: Calculate the total

Now we have:

\[ 1 + 1 + 1 + 1 + 1 + \frac{1}{2} = 5 + \frac{1}{2} = \frac{11}{2} \]

### Final Answer

Thus, the value of the expression is

\[ \frac{11}{2} \text{ or } 5.5 \]