To solve this problem, we need to match each given sine expression to another sine expression with an identical value. We can utilize the properties and symmetry of sine functions in trigonometry to make these matches.
Step 1: Understand sine symmetry properties.
The sine function has a property where [tex]\(\sin(x) = \sin(\pi - x)\)[/tex]. Applying this property lets us find equivalent sine values for angles within a certain range.
Step 2: Apply properties to given angles.
Let's match each given angle with another angle within [tex]\(\pi\)[/tex] that has an identical sine value:
1. [tex]\(\sin \frac{3\pi}{11}\)[/tex]:
- According to the symmetry property, [tex]\(\sin \frac{3\pi}{11} = \sin(\pi - \frac{3\pi}{11}) = \sin(\frac{8\pi}{11})\)[/tex].
2. [tex]\(\sin \frac{4\pi}{11}\)[/tex]:
- According to the symmetry property, [tex]\(\sin \frac{4\pi}{11} = \sin(\pi - \frac{4\pi}{11}) = \sin(\frac{7\pi}{11})\)[/tex].
3. [tex]\(\sin \frac{6\pi}{11}\)[/tex] doesn’t need transformation to find its paired sine.
By this identification, the pairs are:
- [tex]\( \sin \frac{3\pi}{11} \rightarrow \sin \frac{8\pi}{11} \)[/tex]
- [tex]\( \sin \frac{4\pi}{11} \rightarrow \sin \frac{7\pi}{11} \)[/tex]
- [tex]\( \sin \frac{6\pi}{11} \rightarrow \sin \frac{5\pi}{11} \)[/tex]
Thus the correct matching is:
- [tex]\( \sin \frac{3\pi}{11} \rightarrow \sin \frac{8\pi}{11} \)[/tex]
- [tex]\( \sin \frac{4\pi}{11} \rightarrow \sin \frac{7\pi}{11} \)[/tex]
- [tex]\( \sin \frac{6\pi}{11} \rightarrow \sin \frac{5\pi}{11} \)[/tex]
