Answer Key
11.9 Trigonometric Modelling
Pack A — Answers
| # | Question | Answer |
|---|---|---|
| 1 | State the amplitude of $y = 3\sin x$. | 3 |
| 2 | State the period of $y = \sin(Bx)$ when $B = 2$. | $\pi$ |
| 3 | For $y = 3\sin x + 5$, state the maximum and minimum values. | Max 8, Min 2 |
| 4 | For $y = 3\sin x + 5$, find $y(0)$. | 5 |
| 5 | A sinusoid has amplitude 4, mean line $y = 1$, period $2\pi$, no phase shift. Write its equation in the form $y = A\sin(Bx) + C$. | $y = 4\sin x + 1$ |
| 6 | State the period of $y = \cos\!\left(\dfrac{\pi x}{ 6}\right)$. | 12 |
| 7 | State the phase shift of $y = \sin(x - \pi/4)$ relative to $y = \sin x$. | $\dfrac{\pi}{4}$ units right |
| 8 | The depth of water in a harbour rises and falls. State the kind of function (sine, cosine, linear, exponential) that best models this. | Sinusoidal (sine or cosine). |
| 9 | A periodic process has max value 14 and min value 4. State its amplitude and mean (vertical shift) for a $\sin$-based model. | Amplitude 5; mean 9 |
| 10 | Describe the graph of $y = 3\sin(2x)$: amplitude, period, and any vertical shift. | Amplitude 3; period $\pi$; no vertical shift. |
| 11 | For $y = A\sin(Bx) + C$ with $A = 3$, $B = \pi/6$, $C = 4$, state (a) amplitude, (b) period, (c) maximum. | (a) 3; (b) 12; (c) 7 |
| 12 | Solve $3\sin\!\left(\dfrac{\pi t}{6}\right) + 5 = 7$ for the smallest positive $t$. | $t \approx 1.39$ hours |
| 13 | 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. | $d(0) = 5$ m; max 8 m |
| 14 | Using $d(t) = 3\sin(\pi t/6) + 5$, find the first time after midnight when the depth is exactly 8 m. | $t = 3$ hours |
| 15 | 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. | $h(t) = 15 - 12\cos\!\left(\dfrac{\pi t}{2}\right)$ |
| 16 | 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 $t = 6$? | 18°C |
| 17 | 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. | Period 4 s; amplitude 6 |
| 18 | For $y = A\sin(Bx) + C$, the graph oscillates between $y = 2$ and $y = 12$. State $A$ and $C$. | $A = 5$, $C = 7$ |
| 19 | A sinusoid passes through $(0, 4)$ with maximum, has amplitude 3, and period 8. Write its equation in the form $y = A\cos(Bx) + C$. | $y = 3\cos\!\left(\dfrac{\pi x}{4}\right) + 1$ |
| 20 | 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. | 18 = height of centre above ground; 15 = radius of the wheel. |
| 21 | 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. | $t \approx 1.31$ min |
| 22 | Solve $3\sin\!\left(\dfrac{\pi t}{6}\right) + 5 = 7$ for all $t$ in $[0, 12]$. | $t \approx 1.39$ h or $t \approx 4.61$ h |
| 23 | 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. | $L(t) = -3\cos\!\left(\dfrac{\pi t}{6}\right) + 12$ |
| 24 | 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? | (a) $t = 15$ h (3pm); (b) 28°C |
| 25 | 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. | $\approx 5.22$ hours |
| 26 | A sinusoid $y = A\sin(Bx) + C$ has amplitude 4, period 6 and passes through $(0, 2)$. Find $A$ (assuming positive) and $C$. | $A = 4$, $C = 2$ |
| 27 | 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 $A < 0$. | $y = -4\cos\!\left(\dfrac{\pi t}{4}\right) + 6$ |
| 28 | Rewrite $y = \sin x$ in terms of cosine using a phase shift. | $y = \cos(x - \pi/2)$ |
| 29 | For $y = 5\sin(2x) + 8$, find the average value of $y$ over one full period. | 8 |
| 30 | 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 $x(0) = 15$. | $x(t) = 15 \cos(\pi t)$ |
| 31 | 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? | $\approx 1.38$ min |
| 32 | 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. | $t \approx 0.671$ h |
| 33 | A pendulum has displacement at $t = 0$ of 8 cm (max). Period is 2 s. Find (a) amplitude, (b) $B$, (c) the position model $x(t) = A\cos(Bt)$. | (a) 8; (b) $\pi$; (c) $x(t) = 8\cos(\pi t)$ |
| 34 | 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). | $L(t) = -3.5\cos(\pi t/6) + 12$; $L(3) = 12$ h |
| 35 | 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) 60 Hz; (b) $t = \dfrac{1}{240}$ s $\approx 4.17$ ms |
| 36 | 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. | $y(t) = 4\sin\!\left(\dfrac{\pi t}{4}\right)$ |
| 37 | 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. | Periodic with period $2\pi$ (least common period of components). |
| 38 | 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 $A, B, C$. | $A = 4$, $B = \dfrac{\pi}{6}$, $C = 2$ |
| 39 | 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. | $\dfrac{1}{3}$ (8 hours per day) |
| 40 | 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. | 10 W |
Pack B — Answers
| # | Question | Answer |
|---|---|---|
| 1 | State the amplitude of $y = 7\sin x$. | 7 |
| 2 | State the period of $y = \sin(Bx)$ when $B = \pi/3$. | 6 |
| 3 | For $y = 2\sin x + 7$, state the maximum and minimum values. | Max 9, Min 5 |
| 4 | For $y = 2\sin x + 7$, find $y(0)$. | 7 |
| 5 | A sinusoid has amplitude 4, mean line $y = 1$, period $2\pi$, no phase shift. Write its equation in the form $y = A\sin(Bx) + C$. | $y = 6\sin(2x) + 2$ |
| 6 | State the period of $y = \cos\!\left(\dfrac{\pi x}{ 4}\right)$. | 8 |
| 7 | State the phase shift of $y = \sin(x - \pi/4)$ relative to $y = \sin x$. | $\dfrac{\pi}{3}$ units left |
| 8 | The depth of water in a harbour rises and falls. State the kind of function (sine, cosine, linear, exponential) that best models this. | Sinusoidal. |
| 9 | A periodic process has max value 14 and min value 4. State its amplitude and mean (vertical shift) for a $\sin$-based model. | Amplitude 7; mean 15 |
| 10 | Describe the graph of $y = 3\sin(2x)$: amplitude, period, and any vertical shift. | Amplitude 4; period 6; shifted down 1. |
| 11 | For $y = A\sin(Bx) + C$ with $A = 3$, $B = \pi/6$, $C = 4$, state (a) amplitude, (b) period, (c) maximum. | (a) 5; (b) 8; (c) 12 |
| 12 | Solve $3\sin\!\left(\dfrac{\pi t}{6}\right) + 5 = 7$ for the smallest positive $t$. | $t \approx 1.33$ hours |
| 13 | 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. | $d(0) = 6$ m; max 8 m |
| 14 | Using $d(t) = 3\sin(\pi t/6) + 5$, find the first time after midnight when the depth is exactly 8 m. | $t = 9$ hours |
| 15 | 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. | $h(t) = 12 - 10\cos\!\left(\dfrac{\pi t}{3}\right)$ |
| 16 | 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 $t = 6$? | 23°C |
| 17 | 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. | Period 4 s; amplitude 4 |
| 18 | For $y = A\sin(Bx) + C$, the graph oscillates between $y = 2$ and $y = 12$. State $A$ and $C$. | $A = 5$, $C = 2$ |
| 19 | A sinusoid passes through $(0, 4)$ with maximum, has amplitude 3, and period 8. Write its equation in the form $y = A\cos(Bx) + C$. | $y = -4\cos\!\left(\dfrac{\pi x}{3}\right) + 2$ |
| 20 | 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. | 22 = mean (average) temperature; 6 = amplitude of temperature variation. |
| 21 | 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. | $t \approx 1.59$ min |
| 22 | Solve $3\sin\!\left(\dfrac{\pi t}{6}\right) + 5 = 7$ for all $t$ in $[0, 12]$. | $t \approx 1.33$ h or $t \approx 6.67$ h |
| 23 | 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. | $L(t) = -4\cos\!\left(\dfrac{\pi t}{6}\right) + 12$ |
| 24 | 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? | (a) $t = 14$ h (2pm); (b) 24°C |
| 25 | 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. | $\approx 8$ hours |
| 26 | A sinusoid $y = A\sin(Bx) + C$ has amplitude 4, period 6 and passes through $(0, 2)$. Find $A$ (assuming positive) and $C$. | $A = 5$, $C = 7$ |
| 27 | 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 $A < 0$. | $y = -5\cos\!\left(\dfrac{\pi t}{3}\right) + 9$ |
| 28 | Rewrite $y = \sin x$ in terms of cosine using a phase shift. | $y = \sin(x + \pi/2)$ |
| 29 | For $y = 5\sin(2x) + 8$, find the average value of $y$ over one full period. | 12 |
| 30 | 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 $x(0) = 15$. | $x(t) = 20 \cos\!\left(\dfrac{2\pi t}{3}\right)$ |
| 31 | 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? | $\approx 0.823$ min |
| 32 | 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. | $t \approx 1.39$ h |
| 33 | A pendulum has displacement at $t = 0$ of 8 cm (max). Period is 2 s. Find (a) amplitude, (b) $B$, (c) the position model $x(t) = A\cos(Bt)$. | (a) 12; (b) $\pi/2$; (c) $x(t) = 12\cos(\pi t/2)$ |
| 34 | 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). | $L(t) = -3.5\cos(\pi t/6) + 12.5$; $L(3) = 12.5$ h |
| 35 | 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) 50 Hz; (b) $t = \dfrac{1}{200}$ s $= 5$ ms |
| 36 | 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. | $y(t) = 3\sin\!\left(\dfrac{\pi t}{3}\right)$ |
| 37 | 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. | Periodic with period $2\pi$. |
| 38 | 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 $A, B, C$. | $A = 6$, $B = \dfrac{\pi}{4}$, $C = 5$ |
| 39 | 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. | $\approx 1/6$ (about 4 h) |
| 40 | 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. | 18 W |
Problems — Worked Solutions
**Tide model.** The depth $d$ (m) of water in a harbour follows $d(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 5$, where $t$ is hours after midnight. (a) State the amplitude, period, and mean depth. (b) Find the maximum and minimum depths. (c) Find the first time after midnight at which the depth is exactly 7 m, to 3 s.f.
(a) Amplitude 3 m; period 12 h; mean 5 m. (b) Max 8 m; min 2 m. (c) $t \approx 1.39$ h (about 01:24).
**Ferris wheel.** A Ferris wheel of radius 15 m has its centre 18 m above ground. It rotates anti-clockwise with period 4 minutes. A capsule starts at the lowest point at $t = 0$. (a) Explain why $h(t) = 18 - 15\cos\!\left(\dfrac{\pi t}{2}\right)$ models its height. (b) State the max and min heights. (c) Find the first time the capsule is 25 m above ground, to 3 s.f. (d) State, without further calculation, the total time per revolution that the capsule is above 25 m.
(a) See working. (b) Max 33 m, min 3 m. (c) $t \approx 1.31$ min. (d) ≈ 1.38 min.
**Periodic features.** For $y = A\sin(Bx) + C$ with $A = 3$, $B = \dfrac{\pi}{6}$, $C = 4$: (a) State the amplitude. (b) State the period. (c) State the maximum and minimum values of $y$. (d) Sketch $y$ for $0 \leq x \leq 12$, marking the $y$-intercept and the first maximum.
(a) 3. (b) 12. (c) Max 7; min 1. (d) Starts at $(0, 4)$ (mean), rises to max 7 at $x = 3$.
**Daylight in a city.** Daylight varies from 9 h on 21 December ($t = 0$, where $t$ is in months) to 15 h on 21 June ($t = 6$). (a) State the amplitude and the mean. (b) Write a model $L(t) = -A\cos\!\left(\dfrac{\pi t}{6}\right) + D$. (c) Predict the daylight on 21 March ($t = 3$). (d) Find $t$ where the model first gives $L = 13$ h, to 3 s.f.
(a) Amplitude 3; mean 12. (b) $L(t) = -3\cos(\pi t / 6) + 12$. (c) 12 h. (d) $t \approx 4.09$ months.
**Pendulum.** A pendulum swings horizontally and its position from rest follows $x(t) = 10\cos(\pi t)$, where $x$ is cm and $t$ is seconds. (a) State the amplitude and period. (b) Find $x(0)$, $x(0.5)$, $x(1)$. (c) Find the times in $[0, 2]$ s at which $x = 0$. (d) Find the maximum speed (in cm/s) by symbolic differentiation, $v(t) = x'(t)$.
(a) Amplitude 10; period 2. (b) 10, 0, $-10$. (c) $t = 0.5$ or $1.5$. (d) Max speed $10\pi \approx 31.4$ cm/s.
**Convert sin ↔ cos.** Rewrite each function using a single sine OR cosine and a phase shift. (a) $y = \sin x$ as a cosine (b) $y = \cos x$ as a sine (c) $y = -\cos x$ as a cosine with a phase shift
(a) $y = \cos(x - \pi/2)$. (b) $y = \sin(x + \pi/2)$. (c) $y = \cos(x - \pi)$.
**Fitting from a table.** Data observed: | $t$ (s) | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---| | $y$ | 5 | 9 | 5 | 1 | 5 | 9 | (a) State the period. (b) State the amplitude and mean. (c) Write a model $y(t) = A\sin(Bt) + C$ given that $y(0)$ is at the mean and rising.
(a) 4 s. (b) Amplitude 4; mean 5. (c) $y(t) = 4\sin(\pi t / 2) + 5$.
**Daylight + threshold.** Using $L(t) = -3\cos(\pi t / 6) + 12$ (h, $t$ in months from 21 Dec): (a) For how many months per year is $L > 13$ h? (b) State the start and end times of this interval, to 3 s.f. (c) Sketch $L$ for $0 \leq t \leq 12$ marking $L = 13$ on the $y$-axis.
(a) $\approx 3.83$ months. (b) From $t \approx 4.09$ to $t \approx 7.91$ months. (c) Sketch.
**Periodic vs non-periodic.** Decide whether each function is periodic. Justify, and (where periodic) state the period. (a) $y = \sin x + \cos x$ (b) $y = \sin x + x$ (c) $y = \sin(2x) + \sin(3x)$ [EXT] (d) $y = \sin x \cdot \cos x$
(a) Periodic, period $2\pi$. (b) Not periodic — the $x$ term grows. (c) Periodic, period $2\pi$. (d) Periodic, period $\pi$ (since $\sin x \cos x = \frac{1}{2}\sin 2x$).
**Modelling — sound wave.** A sound wave is modelled by $P(t) = 0.5\sin(2\pi \cdot 440 \cdot t)$, where $P$ is pressure in Pa and $t$ in seconds. (a) State the amplitude and frequency. (b) Find the period. (c) Find $P(0)$, $P(1/1760)$. (d) Comment on what the frequency 440 Hz represents musically.
(a) Amplitude 0.5 Pa; frequency 440 Hz. (b) Period $\approx 2.27$ ms. (c) $P(0) = 0$; $P(1/1760) = 0.5$. (d) The pitch A4 (concert A).
**Inverse modelling.** A tidal model gives $d(t) = 4\sin\!\left(\dfrac{\pi t}{6}\right) + 7$ where $t$ is hours since the previous high tide (so $d$ peaks at $t = 3$). (a) State the amplitude, period, and average depth. (b) Find the depth at $t = 0$. (c) Find the first $t > 0$ at which the depth is again 7 m. (d) Find the duration of one "low-tide window" defined as $d < 5$.
(a) Amplitude 4, period 12, mean 7. (b) $d(0) = 7$. (c) $t = 6$ h. (d) $\approx 4.40$ h.
**Investigation — fit & predict.** Population of foxes in a region is observed monthly: | $t$ | 1 | 4 | 7 | 10 | |---|---|---|---|---| | $P$ | 410 | 240 | 410 | 580 | (a) Argue that a sinusoidal model is reasonable. (b) Estimate amplitude, mean, and period from the data. (c) Fit a model $P(t) = A\sin(B(t - C)) + D$. (d) Use the model to predict the population at $t = 12$.
(a) Data oscillates: high–low–high–even higher. The change is broadly periodic. (b) Amplitude $\approx 170$; mean $\approx 410$; period $\approx 12$. (c) Approximately $P(t) = 170\sin\!\left(\dfrac{\pi(t - 7)}{6}\right) + 410$. (d) $P(12) \approx 410 + 170\sin(5\pi/6) = 410 + 85 = 495$.