To solve for [tex]\( y = \sin (\sin (x)) \)[/tex] at the point [tex]\( x = 3\pi \)[/tex], we need to follow these steps:
1. Understand the Inner Function:
- First, recognize the inner function: [tex]\( \sin(x) \)[/tex].
2. Evaluate the Inner Function at [tex]\( x = 3\pi \)[/tex]:
- Calculate [tex]\( \sin(3\pi) \)[/tex].
- Remember that [tex]\( \sin(\theta) \)[/tex] is a periodic function with a period of [tex]\( 2\pi \)[/tex]. Therefore, [tex]\( \sin(3\pi) \)[/tex] can be simplified as follows:
[tex]\[
\sin(3\pi) = \sin(\pi + 2\pi) = \sin(\pi)
\][/tex]
- We know that [tex]\( \sin(\pi) = 0 \)[/tex].
3. Use the Result from the Inner Function:
- Now that we have [tex]\( \sin(3\pi) = 0 \)[/tex], we substitute this result into the outer function: [tex]\( \sin (\sin(3\pi)) \)[/tex].
4. Evaluate the Outer Function:
- This simplifies to [tex]\( \sin(0) \)[/tex].
- We know that [tex]\( \sin(0) = 0 \)[/tex].
5. Therefore:
- [tex]\( \sin(\sin(3\pi)) = \sin(0) = 0 \)[/tex].
So, the detailed solution shows that the value of [tex]\( y \)[/tex] at [tex]\( x = 3\pi \)[/tex] for the function [tex]\( y = \sin(\sin(x)) \)[/tex] is indeed [tex]\( 0 \)[/tex].
