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Find an equation for a sinusoidal function that has period $4\pi$ , amplitude $2$ , and contains the point $(2\pi,4)$ . Write your answer in the form $f(x)=A\sin(Bx+C)+D$ , where $A$ , $B$ , $C$ , and $D$ are real numbers.

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Write equations of sine and cosine functions using properties

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Q. Find an equation for a sinusoidal function that has period

4\pi

, amplitude

2

, and contains the point

(2\pi,4)

. Write your answer in the form

f(x)=A\sin(Bx+C)+D

, where

A

,

B

,

C

, and

D

are real numbers.

Determine B value: Determine the value of B using the period of the sinusoidal function. $\newline$ The period of a sinusoidal function in the form $f(x) = A\sin(Bx + C) + D$ is given by $(2\pi)/B$ . Since the period is given as $4\pi$ , we can set up the equation: $\newline$ $4\pi = (2\pi)/B$ $\newline$ Solving for B gives us: $\newline$ $B = (2\pi)/(4\pi)$ $\newline$ $B = 1/2$
Write general form: Write the general form of the sinusoidal function using the amplitude $A$ and the value of $B$ we just found. $\newline$ The amplitude is given as $2$ , so $A = 2$ . We have found $B = \frac{1}{2}$ . The general form of the function so far is: $\newline$ $f(x) = 2\sin(\left(\frac{1}{2}\right)x + C) + D$
Use given point: Use the given point $(2\pi, 4)$ to find the values of $C$ and $D$ . Plugging in the point $(2\pi, 4)$ into the equation gives us: $4 = 2\sin\left(\frac{1}{2}(2\pi) + C\right) + D$ Simplifying the sine term: $4 = 2\sin(\pi + C) + D$ Since $\sin(\pi + C) = \sin(\pi)\cos(C) + \cos(\pi)\sin(C) = 0\cdot\cos(C) - 1\cdot\sin(C) = -\sin(C)$ , the equation becomes: $4 = -2\sin(C) + D$
Determine $C$ value: Determine the value of $C$ using the properties of the sine function. $\newline$ Since the sine function oscillates between $-1$ and $1$ , and the coefficient of the sine function is $-2$ , the only way for $\sin(C)$ to be $0$ is for $C$ to be an integer multiple of $\pi$ . Therefore, we can choose $C = 0$ (as one of the simplest solutions) without loss of generality.
Solve for D: Solve for D using the value of $C$ we just determined. $\newline$ Substituting $C = 0$ into the equation from Step $3$ : $\newline$ $4 = -2\sin(0) + D$ $\newline$ $4 = 0 + D$ $\newline$ $D = 4$
Write final equation: Write the final equation of the sinusoidal function using the values of $A$ , $B$ , $C$ , and $D$ we have found. $\newline$ Substituting $A = 2$ , $B = \frac{1}{2}$ , $C = 0$ , and $D = 4$ into the general form: $\newline$ $f(x) = 2\sin(\left(\frac{1}{2}\right)x + 0) + 4$ $\newline$ Simplifying the equation: $\newline$ $f(x) = 2\sin(\left(\frac{1}{2}\right)x) + 4$

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