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Find an equation for a sinusoidal function that has period $2\pi$ , amplitude $2$ , and contains the point $\left(\frac{3\pi}{2}, 0\right)$ . Write your answer in the form $f(x)=A\cos(Bx+C)+D$ , 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

2

, and contains the point

\left(\frac{3\pi}{2}, 0\right)

. Write your answer in the form

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

, where

A

,

B

,

C

, and

D

are real numbers.

Determine B Value: Determine the value of B using the given period of the function. $\newline$ Since the period of a cosine function is given by $(2\pi)/B$ , we can set this equal to the given period of $2\pi$ and solve for B. $\newline$ Period: $(2\pi)/B = 2\pi$ $\newline$ $B = (2\pi)/(2\pi)$ $\newline$ $B = 1$
Set Amplitude and B: Since the amplitude is given as $2$ , we know that $A = 2$ . So far, we have: $\newline$ $f(x) = 2\cos(Bx + C) + D$ $\newline$ Substituting $B = 1$ , we get: $\newline$ $f(x) = 2\cos(x + C) + D$
Find C and D Values: Next, we need to determine the values of C and D. Since the function contains the point $(3\pi/2, 0)$ , we can substitute $x = 3\pi/2$ and $f(x) = 0$ into the equation and solve for C and D. $\newline$ $0 = 2\cos(3\pi/2 + C) + D$ $\newline$ Since $\cos(3\pi/2) = 0$ , the equation simplifies to: $\newline$ $0 = 2(0) + D$ $\newline$ $D = 0$
Determine Phase Shift: Now we need to find the value of $C$ . We know that the cosine function has a value of $0$ at $\frac{3\pi}{2}$ when it is at the mid-point between its maximum and minimum values. Since the amplitude is $2$ , the maximum and minimum values are $2$ and $-2$ , respectively. Therefore, the cosine function must be at $\frac{3\pi}{2}$ for the first time at the mid-point of its period, which is $\pi$ . This means that the phase shift $C$ must be such that $x + C = \pi$ when $0$ $0$ . $\newline$ $0$ $1$ $\newline$ $0$ $2$ $\newline$ $0$ $3$
Substitute Values for Final Equation: Substitute the values of $A$ , $B$ , $C$ , and $D$ into the general form of the equation to get the final equation of the sinusoidal function. $\newline$ $f(x) = 2\cos(x - \frac{\pi}{2}) + 0$ $\newline$ $f(x) = 2\cos(x + \frac{\pi}{2})$ (since cosine is an even function, $\cos(-\theta) = \cos(\theta)$ )

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