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Find an equation for a sinusoidal function that has period $2\pi$ , amplitude $1$ , and contains the point $(0,0)$ . 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

2\pi

, amplitude

1

, and contains the point

(0,0)

. Write your answer in the form

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

, where

A

,

B

,

C

, and

D

are real numbers.

Find B: We know: Period = $2\pi$ $\newline$ Find the value of B. $\newline$ Period: $(2\pi)/B$ $\newline$ $2\pi = (2\pi)/B$ $\newline$ $B = (2\pi)/(2\pi)$ $\newline$ $B = 1$
Function with $A=1$ : Since the amplitude is $1$ , $A = 1$ . Now we have: $f(x) = 1\sin(1x + C) + D$
Find D: We need to find the values of $C$ and $D$ . Since the function contains the point $(0,0)$ , we can plug in $x = 0$ and $f(x) = 0$ to find $D$ . $\newline$ $f(0) = 1\sin(1\cdot 0 + C) + D$ $\newline$ $0 = 1\sin(C) + D$
Choose $C=0$ : Since $\sin(C)$ can range from $-1$ to $1$ , and we want the function to pass through $(0,0)$ , the most straightforward choice is to make $\sin(C) = 0$ . This happens when $C$ is an integer multiple of $\pi$ . To keep the function simple, we choose $C = 0$ .

$0 = 1\sin(0) + D$
$\sin(C)$ $0$
$\sin(C)$ $1$
Write in Amplitude-Phase Form: We found: $\newline$ $A = 1$ $\newline$ $B = 1$ $\newline$ $C = 0$ $\newline$ $D = 0$ $\newline$ Write the equation in amplitude-phase form. $\newline$ Substitute values of $A$ , $B$ , $C$ , and $D$ $\newline$ $f(x) = 1\sin(1x + 0) + 0$ $\newline$ $= \sin(x)$

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