To match the trigonometric functions that have the same value, we need to make use of the trigonometric identity for sine functions: [tex]\( \sin(x) = \sin(\pi - x) \)[/tex].
This identity tells us that sin(x) has the same value as sin(π - x).
Let's apply this property step-by-step:
1. Pair involving [tex]\( \sin \frac{\pi}{11} \)[/tex]:
- We need to find [tex]\( \pi - \frac{\pi}{11} \)[/tex]:
[tex]\[
\pi - \frac{\pi}{11} = \frac{11\pi}{11} - \frac{\pi}{11} = \frac{10\pi}{11}
\][/tex]
- Therefore, [tex]\( \sin \frac{\pi}{11} = \sin \frac{10\pi}{11} \)[/tex].
2. Pair involving [tex]\( \sin \frac{3\pi}{11} \)[/tex]:
- We need to find [tex]\( \pi - \frac{3\pi}{11} \)[/tex]:
[tex]\[
\pi - \frac{3\pi}{11} = \frac{11\pi}{11} - \frac{3\pi}{11} = \frac{8\pi}{11}
\][/tex]
- Therefore, [tex]\( \sin \frac{3\pi}{11} = \sin \frac{8\pi}{11} \)[/tex].
3. Pair involving [tex]\( \sin \frac{5\pi}{11} \)[/tex]:
- We need to find [tex]\( \pi - \frac{5\pi}{11} \)[/tex]:
[tex]\[
\pi - \frac{5\pi}{11} = \frac{11\pi}{11} - \frac{5\pi}{11} = \frac{6\pi}{11}
\][/tex]
- Therefore, [tex]\( \sin \frac{5\pi}{11} = \sin \frac{6\pi}{11} \)[/tex].
Now, let's compile the matched pairs:
- [tex]\( \sin \frac{\pi}{11} \)[/tex] is paired with [tex]\( \sin \frac{10\pi}{11} \)[/tex]
- [tex]\( \sin \frac{3\pi}{11} \)[/tex] is paired with [tex]\( \sin \frac{8\pi}{11} \)[/tex]
- [tex]\( \sin \frac{5\pi}{11} \)[/tex] is paired with [tex]\( \sin \frac{6\pi}{11} \)[/tex]
Using the trigonometric function identities, you've managed to match these pairs correctly. Here are the pairs in a tabulated format for better clarity:
[tex]\[
\begin{array}{c|c}
\text{Trigonometric Function 1} & \text{Trigonometric Function 2} \\
\hline
\sin \frac{\pi}{11} & \sin \frac{10\pi}{11} \\
\sin \frac{3\pi}{11} & \sin \frac{8\pi}{11} \\
\sin \frac{5\pi}{11} & \sin \frac{6\pi}{11}
\end{array}
\][/tex]
