11.9 Trigonometric Modelling

**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.

**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.

**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.

**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.

**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, $v(t) = x'(t)$.

**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

**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.

**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.

**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) $y = \sin x \cdot \cos x$

**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.

**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 $d < 5$.

**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 $t = 12$.