In general a trigonometric function of the form:
[tex]y=A\sin (B(x+C))+D[/tex]has the following properties:
• A is the amplitude.
,• C is the phase shift (positive to the left)
,• D is the vertical shift
,• The period is given as:
[tex]\frac{2\pi}{B}[/tex]
From the information given we have that the amplitude is 3, then:
[tex]A=3[/tex]The period is 6pi then we have:
[tex]\begin{gathered} 6\pi=\frac{2\pi}{B} \\ B=\frac{2\pi}{6\pi} \\ B=\frac{1}{3} \end{gathered}[/tex]The horizontal shift is:
[tex]C=-\frac{3\pi}{2}[/tex]And the vertical shift is:
[tex]D=-1[/tex]Once we know the values we plug them in the general expression for the sine function, our function is:
[tex]y=3\sin (\frac{1}{3}(x-\frac{3\pi}{2}))-1[/tex]Now that we have the function we can find the its value when x=2pi, plugging this value of x in the expression we have:
[tex]\begin{gathered} y=3\sin (\frac{1}{3}(2\pi-\frac{3\pi}{2}))-1 \\ y=3\sin (\frac{1}{3}(\frac{\pi}{2}))-1 \\ y=3\sin (\frac{\pi}{6})=-1 \\ y=3(\frac{1}{2})-1 \\ y=\frac{3}{2}-1 \\ y=\frac{1}{2} \end{gathered}[/tex]Therefore, the value of y for the given x is 1/2