The value of log `sin0^(@)+log sin1^(@)+log sin2^(@)+* * *+log sin 90^(@)`is ________.

Options

A. 0
B. 1
C. -1
D. undefined

Explanation

To find the value of \( \log \sin 0^\circ + \log \sin 1^\circ + \log \sin 2^\circ + \ldots + \log \sin 90^\circ \), we can follow these steps:

### Step 1: Use the property of logarithms
We know that \( \log a + \log b = \log(ab) \). Therefore, we can combine the logarithmic terms:

\[ \log \sin 0^\circ + \log \sin 1^\circ + \log \sin 2^\circ + \ldots + \log \sin 90^\circ = \log(\sin 0^\circ \cdot \sin 1^\circ \cdot \sin 2^\circ \cdots \sin 90^\circ) \]

### Step 2: Evaluate \( \sin 0^\circ \)
We know that \( \sin 0^\circ = 0 \). Therefore, when we multiply \( \sin 0^\circ \) with any other terms, the product will be zero:

\[ \sin 0^\circ \cdot \sin 1^\circ \cdot \sin 2^\circ \cdots \sin 90^\circ = 0 \]

### Step 3: Substitute back into the logarithm
Now substituting back into our logarithmic expression, we have:

\[ \log(0) \]

### Step 4: Determine the value of \( \log(0) \)
The logarithm of zero is undefined. Therefore, we conclude that:

\[ \log \sin 0^\circ + \log \sin 1^\circ + \log \sin 2^\circ + \ldots + \log \sin 90^\circ \text{ is undefined.} \]

### Final Answer
The value of \( \log \sin 0^\circ + \log \sin 1^\circ + \log \sin 2^\circ + \ldots + \log \sin 90^\circ \) is **undefined**.