`sin163^@ cos347^@+sin73^@sin167^@=`

Options

A. 0
B. `1/2`
C. 1
D. `2/3`

Explanation

To solve the expression \( \sin 163^\circ \cos 347^\circ + \sin 73^\circ \sin 167^\circ \), we can simplify each term using trigonometric identities.

### Step-by-step Solution:

1. **Rewrite the angles using identities**:
- \( \sin 163^\circ = \sin(180^\circ - 17^\circ) = \sin 17^\circ \)
- \( \cos 347^\circ = \cos(360^\circ - 13^\circ) = \cos 13^\circ \)
- \( \sin 73^\circ = \sin(90^\circ - 17^\circ) = \cos 17^\circ \)
- \( \sin 167^\circ = \sin(180^\circ - 13^\circ) = \sin 13^\circ \)

2. **Substitute the rewritten angles into the expression**:
\[ \sin 163^\circ \cos 347^\circ + \sin 73^\circ \sin 167^\circ = \sin 17^\circ \cos 13^\circ + \cos 17^\circ \sin 13^\circ \]

3. **Use the sine addition formula**:
The sine addition formula states that:
\[ \sin A \cos B + \cos A \sin B = \sin(A + B) \]
Here, let \( A = 17^\circ \) and \( B = 13^\circ \):
\[ \sin 17^\circ \cos 13^\circ + \cos 17^\circ \sin 13^\circ = \sin(17^\circ + 13^\circ) = \sin 30^\circ \]

4. **Calculate \( \sin 30^\circ \)**:
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]

5. **Final Result**:
Therefore, the value of the original expression is:
\[ \sin 163^\circ \cos 347^\circ + \sin 73^\circ \sin 167^\circ = \frac{1}{2} \]