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Question 1

In radians, write $y=A\sin\bigl(B(x - C)\bigr)+D$ for a sinusoid with amplitude $5$ , period $\frac{2\pi}{3}$ , shifted $\frac{\pi}{4}$ to the left, and midline $y=1$ .

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Question 2

Find a model $y=A\sin(Bx)+D$ with amplitude $3$ , period $180^\circ$ , and midline $y=2$ , assuming no horizontal shift.

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Question 3

A daily temperature oscillation has amplitude $10^{\circ}\text{C}$ , average $20^{\circ}\text{C}$ , and peaks at 3 pm. Model the temperature $T(t)$ in the form $T=10\sin\bigl(B(t - C)\bigr)+20$ , where $t$ is hours after midnight.

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Question 4

A mass on a spring undergoes simple harmonic motion described by $d(t) = A\sin(\omega t + \phi)$ . If the amplitude is $3\text{ cm}$ , the period is $4\text{ s}$ , and the mass passes upward through equilibrium at $t = 1\text{ s}$ , find $d(t)$ .

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Question 5

The height of the tide in a harbour, $h$ metres, can be modelled by the function $h(t) = 2\sin\bigl(B(t - C)\bigr) + 5$ where $t$ is the time in hours after midnight. The tide has an amplitude of $2$ metres, a mean level of $5$ metres, and a period of $12$ hours. A high tide occurs at $t = 6$ .

Find the expression for $h(t)$ by determining the values of $B$ and $C$ .

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Question 6

A Ferris wheel has radius $20$ m, with its centre $22$ m above the ground, and completes one revolution per minute. A passenger starts at the lowest point at $t=0$ . Write the height $H(t)$ in metres at time $t$ minutes in the form $H(t)=20\sin(Bt+\phi)+22$ .

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Question 7

Model the average daylight hours $D(d)$ over a year with amplitude $2$ , mean $12$ , maximum on day $172$ , and period $365$ days, in the form $D=2\sin\left(\frac{2\pi}{365}(d - C)\right)+12$ .

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Question 8

Write a model $y = A \sin(Bx + C) + D$ for a sinusoid with amplitude $2$ , period $60^\circ$ , midline $y = 0$ , and shifted $15^\circ$ to the left.

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Question 9

Determine the sinusoidal model $y= A\sin\bigl(B(x - C)\bigr)+D$ with amplitude $4$ , period $120^\circ$ , midline $y=-1$ , and phase shift $30^\circ$ to the right.

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Question 10

A sinusoid has a maximum value of $8$ at $x = 10^\circ$ and a minimum value of $2$ at $x = 100^\circ$ , repeating every $180^\circ$ . Find its equation in the form $y = A \sin(B(x - C)) + D$ .

Find the equation of the sinusoid in the form $y = A \sin(B(x - C)) + D$ .

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Question 11

Determine the equation $y=A\sin\bigl(B(x - C)\bigr)+D$ for a sinusoid with amplitude $6$ , period $100^\circ$ , midline $y=-2$ , phase shift $20^\circ$ right, and reflected across its midline.

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Question 12

A sinusoidal model of the form $y = A\sin(Bx)+D$ is defined by its amplitude, period, and principal axis.

Formulate a sinusoidal model of the form $y = A\sin(Bx)+D$ that has amplitude $5$ , period $90^\circ$ , and principal axis $y=11$ , with no phase shift.

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Question Type 15: Formulating sinusoidal models given descriptions Exercises