The vertical displacement, d(t) in centimeters, of a particle executing simple harmonic mo… — ACT

The vertical displacement, $d(t)$ in centimeters, of a particle executing simple harmonic motion is modeled by the trigonometric function $d(t) = A \sin(B(t - C)) + D$ , where $A > 0$ , $B > 0$ , and $C$ represents the smallest non-negative phase shift in seconds. The graph of $d(t)$ completes one full cycle every $\frac{2\pi}{3}$ seconds, has a maximum value of $7\text{ cm}$ at $t = \frac{5\pi}{18}\text{ seconds}$ , and has a minimum value of $-3\text{ cm}$ . What is the value of $C$ ?

A
$\frac{\pi}{18}$
$\frac{\pi}{9}$ Answer
C
$\frac{2\pi}{9}$
D
$\frac{\pi}{3}$
E
$\frac{5\pi}{18}$

Answer

\frac{\pi}{9}

To find the phase shift

C

, first determine

B

using the period formula

\text{Period} = \frac{2\pi}{B}

. Since the period is

\frac{2\pi}{3}

B = 3

. The function achieves a maximum when its sine argument equals

\frac{\pi}{2} + 2k\pi

. Setting

3\left(\frac{5\pi}{18} - C\right) = \frac{\pi}{2}

yields

\frac{5\pi}{6} - 3C = \frac{\pi}{2}

, which simplifies to

3C = \frac{\pi}{3}

, giving

C = \frac{\pi}{9}

Step-by-Step Solution

Determine the value of

B

from the period of the function.

B = 3

The standard period for a sine function is

2\pi

. Given that the period is

\frac{2\pi}{3}

, we set

\frac{2\pi}{B} = \frac{2\pi}{3}

, which yields

B = 3

Determine the maximum value equation for the parent sine function.

Argument equals

\frac{\pi}{2}

The standard sine function

\sin(\theta)

achieves its first positive maximum at

\theta = \frac{\pi}{2}

. Thus, for

d(t)

, the maximum occurs when

B(t - C) = \frac{\pi}{2}

Substitute

B = 3

and

t = \frac{5\pi}{18}

into the argument equation and solve for

C

C = \frac{\pi}{9}

Substitute the given values:

3\left(\frac{5\pi}{18} - C\right) = \frac{\pi}{2} \implies \frac{5\pi}{6} - 3C = \frac{\pi}{2}

. Subtract

\frac{5\pi}{6}

from both sides to get

-3C = \frac{3\pi}{6} - \frac{5\pi}{6} = -\frac{2\pi}{6} = -\frac{\pi}{3}

. Dividing by

-3

gives

C = \frac{\pi}{9}

Key Concept

Phase shift and parameter identification from graphs of transformed sine functions

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