11.9 Trigonometric Modelling

State the amplitude of $y = 3\sin x$.

State the period of $y = \sin(Bx)$ when $B = 2$.

For $y = 3\sin x + 5$, state the maximum and minimum values.

For $y = 3\sin x + 5$, find $y(0)$.

A sinusoid has amplitude 4, mean line $y = 1$, period $2\pi$, no phase shift. Write its equation in the form $y = A\sin(Bx) + C$.

State the period of $y = \cos\!\left(\dfrac{\pi x}{ 6}\right)$.

State the phase shift of $y = \sin(x - \pi/4)$ relative to $y = \sin x$.

The depth of water in a harbour rises and falls. State the kind of function (sine, cosine, linear, exponential) that best models this.

A periodic process has max value 14 and min value 4. State its amplitude and mean (vertical shift) for a $\sin$-based model.

Describe the graph of $y = 3\sin(2x)$: amplitude, period, and any vertical shift.

For $y = A\sin(Bx) + C$ with $A = 3$, $B = \pi/6$, $C = 4$, state (a) amplitude, (b) period, (c) maximum.

Solve $3\sin\!\left(\dfrac{\pi t}{6}\right) + 5 = 7$ for the smallest positive $t$.

The depth (m) of water in a harbour is $d(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 5$ ($t$ in hours after midnight). Find $d(0)$ and the maximum depth.

Using $d(t) = 3\sin(\pi t/6) + 5$, find the first time after midnight when the depth is exactly 8 m.

A Ferris wheel has radius 12 m and centre 15 m above ground. The lowest point is at $t = 0$. Write a height function $h(t)$ if the period is 4 minutes.

A temperature in a city is modelled by $T(t) = 5\sin\!\left(\dfrac{\pi(t - 6)}{12}\right) + 18$ ($T$ in °C, $t$ in hours after midnight). What is the temperature at $t = 6$?

A periodic process has values 12, 18, 12, 6, 12, 18 at $t = 0, 1, 2, 3, 4, 5$ seconds. State the period and amplitude.

For $y = A\sin(Bx) + C$, the graph oscillates between $y = 2$ and $y = 12$. State $A$ and $C$.

A sinusoid passes through $(0, 4)$ with maximum, has amplitude 3, and period 8. Write its equation in the form $y = A\cos(Bx) + C$.

In $h(t) = -15\cos\!\left(\dfrac{\pi t}{2}\right) + 18$, state the meaning of the values 18 and 15 in the context of a Ferris wheel.

A Ferris wheel of radius 15 m has its centre 18 m above ground. Period 4 minutes; starts at the lowest point. Find the first time (to 3 s.f.) the capsule is 25 m above ground.

Solve $3\sin\!\left(\dfrac{\pi t}{6}\right) + 5 = 7$ for all $t$ in $[0, 12]$.

Daylight in a city varies sinusoidally between 9 h (winter solstice) and 15 h (summer solstice). Write a model $L(t) = A\cos(B(t - C)) + D$ where $t$ is months after January 1.

The temperature in a town is $T(t) = 8\sin\!\left(\dfrac{\pi(t - 9)}{12}\right) + 20$ for $t$ in hours. (a) When is the temperature highest? (b) What is the highest temperature?

The depth model $d(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 5$ (m) for the harbour: for how many hours per cycle is the depth above 7 m? To 3 s.f.

A sinusoid $y = A\sin(Bx) + C$ has amplitude 4, period 6 and passes through $(0, 2)$. Find $A$ (assuming positive) and $C$.

A periodic process has minimum at $t = 0$, max value 10 and min value 2, period 8. Write a model in the form $y = A\cos(Bt) + C$ with $A < 0$.

Rewrite $y = \sin x$ in terms of cosine using a phase shift.

For $y = 5\sin(2x) + 8$, find the average value of $y$ over one full period.

A pendulum swings between $-15$ cm and $15$ cm from rest, with period 2 s. Write a position model $x(t) = A\cos(Bt)$ assuming $x(0) = 15$.

Use the Ferris wheel $h(t) = 18 - 15\cos\!\left(\dfrac{\pi t}{2}\right)$. For how long during one revolution is the capsule above 25 m?

The depth model $d(t) = 3\sin(\pi t/6) + 5$ — find the first time after midnight that the tide is rising and depth reaches 6 m.

A pendulum has displacement at $t = 0$ of 8 cm (max). Period is 2 s. Find (a) amplitude, (b) $B$, (c) the position model $x(t) = A\cos(Bt)$.

Daylight in a town varies from 8.5 h (winter, $t = 0$) to 15.5 h (summer, $t = 6$ months). Write a model $L(t) = A\cos(Bt) + D$, and use it to predict $L(3)$ (i.e. at the spring equinox).

A signal is modelled by $V(t) = 5\sin(120\pi t)$ volts. Find (a) the frequency in Hz, (b) the time of the first peak after $t = 0$.

A motorboat's vertical bob is observed: at $t = 0$ it is at the **mean** height and rising, reaches max at $t = 2$ s. Write a model $y(t) = A\sin(Bt)$ with $A > 0$ and amplitude 4.

The function $f(t)$ describes a combined model: $f(t) = 2\sin(t) + \sin(2t)$. State whether $f$ is periodic; if so, give its period.

A sinusoid $y = A\sin(Bt) + C$ with $A, B > 0$ passes through $(0, 2)$ and reaches its first maximum value 6 at $t = 3$. Find $A, B, C$.

For $T(t) = 6\sin(\pi t/12) + 18$ (°C, $t$ in hours, daily cycle 24 h), find the fraction of one day during which the temperature exceeds 21°C.

A voltage $V(t) = 10\sin(120\pi t)$ V drives a 5 Ω resistor. The average power is $\frac{V_{\text{rms}}^2}{R}$ where $V_{\text{rms}} = \frac{|A|}{\sqrt{2}}$ for a sinusoid. Find the average power, exact and to 3 s.f.