To find the correct sine equation given the specifications: amplitude of 1, period of [tex]\(12 \pi\)[/tex], vertical shift of -5, and horizontal shift of [tex]\(2 \pi\)[/tex], we need to construct the equation of the form:
[tex]\[ y = A \sin(B \theta + C) + D \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude,
- [tex]\( B \)[/tex] is determined by the period using the relationship [tex]\( B = \frac{2\pi}{\text{Period}} \)[/tex],
- [tex]\( C \)[/tex] is the horizontal shift,
- [tex]\( D \)[/tex] is the vertical shift.
Let's identify each of these components step by step:
1. Amplitude ([tex]\(A\)[/tex]):
The amplitude is given as 1. So, [tex]\( A = 1 \)[/tex].
2. Period ([tex]\(T\)[/tex]):
The period is given as [tex]\( 12 \pi \)[/tex]. The relationship between the period and [tex]\( B \)[/tex] is:
[tex]\[ T = \frac{2\pi}{B} \][/tex]
Therefore,
[tex]\[ B = \frac{2\pi}{12\pi} = \frac{1}{6}. \][/tex]
3. Horizontal Shift ([tex]\(C\)[/tex]):
The horizontal shift is given as [tex]\( 2 \pi \)[/tex]. In the formula [tex]\( y = A \sin(B \theta + C) + D \)[/tex], we directly use this shift value. So, [tex]\( C = 2 \pi \)[/tex].
4. Vertical Shift ([tex]\(D\)[/tex]):
The vertical shift is given as -5. Therefore, [tex]\( D = -5 \)[/tex].
Plugging all these values into the equation [tex]\( y = A \sin(B \theta + C) + D \)[/tex]:
[tex]\[ y = 1 \sin\left(\frac{1}{6} \theta + 2 \pi \right) - 5 \][/tex]
Simplifying to match one of the given choices, we have:
[tex]\[ y = \sin\left(\frac{\theta}{6} + 2 \pi \right) - 5 \][/tex]
Comparing this to the options provided:
A. [tex]\( y = \sin \left(\frac{\theta}{6} + 2 \pi \right) - 5 \)[/tex]
B. [tex]\( y = \sin \left(\frac{\theta}{6} + 2 \pi \right) + 5 \)[/tex]
C. [tex]\( y = \sin \left(\frac{\theta}{3} + \frac{\pi}{6} \right) - 5 \)[/tex]
D. [tex]\( y = \sin \left(\frac{\theta}{6} + \frac{\pi}{3} \right) - 5 \)[/tex]
The correct answer is:
[tex]\[ \boxed{A} \][/tex]
