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Trigonometric modelling
Anything that turns, swings or breathes in cycles is trigonometry with its sleeves rolled up. This lesson reads real periodic models, centre line first and amplitude second, and lets the harmonic form tame the messy ones that carry two trig terms at once.
Builds on Compound angles and the harmonic form and Functions in modelling.
Where it earns its keep: Simple harmonic motion on InkPhysics.
IN THIS TOPIC
- Build and read wheel-style models of the form k − R cos ωt, in radians.
- Extract centre, amplitude, period and phase from any periodic model.
- Use the harmonic form to analyse a model carrying two trig terms.
COMMON MISCONCEPTION
A bigger amplitude means a faster oscillation.
A seat on a wheel
A point on a turning wheel is the cleanest periodic model there is. Put the axle at height k, give the wheel radius R and angular speed ω radians per second, and a seat starting at the bottom has height h = k − R cos ωt. Cosine starts at 1, so the model starts at its minimum, which is where the boarding platform is.
WORKED EXAMPLE
Reading the wheel
A wheel's seat height is modelled by h = 12 − 10 cos (πt/15), in metres and seconds. Find the period, the greatest and least heights, and the first time the seat is 17 m up.
The period is 2π ÷ (π/15) = 30 s, and h runs from 12 − 10 = 2 m up to 12 + 10 = 22 m.
Setting h = 17 gives cos (πt/15) = −½, so πt/15 = 2π/3 and t = 10 s.
Every number in that formula is a physical fact about the wheel. The axle sets the centre line, the radius sets the amplitude, the coefficient of t sets the period, and examiners test each one separately.
Two of those dials are easily muddled. Amplitude controls how far the oscillation swings and the period controls how often it repeats, and the wheel shows them living in different parts of the formula, R out in front of the cosine and ω tucked inside it.
Two terms, one tide
Real data often arrives as a sine term plus a cosine term. Last lesson's harmonic form is what makes such a model readable, collapsing the pair into a single wave with a visible amplitude and phase.
GUIDED PRACTICE
Collapse, then read
Water depth is modelled by d = 6 + 2 sin (0.5t) + 1.5 cos (0.5t), in metres and hours. Find the greatest and least depths, and the first time of high water, before opening the working.
Show the working
The trig part folds. R = √(4 + 2.25) = 2.5 and tan α = 1.5/2 gives α = 0.644, so d = 6 + 2.5 sin (0.5t + 0.644).
Depth runs from 6 − 2.5 = 3.5 m to 6 + 2.5 = 8.5 m.
High water needs 0.5t + 0.644 = π/2, so t = 1.85 hours, and every 4π hours after that.
The constant 6 never joined the fold. Harmonic form applies to the oscillating part alone, about whatever centre line the model happens to carry.
INDEPENDENT PRACTICE
Criticise the daylight model
Hours of daylight are modelled by D = 12 + 4.5 sin (2πt/365), with t in days after the spring equinox. State the period and the longest day, and give one reason the model needs refining for a real town.
Show the working
The period is 2π ÷ (2π/365) = 365 days, and the longest day is 12 + 4.5 = 16.5 hours, a quarter of a period in.
One honest criticism: amplitude depends on latitude, so a single fixed 4.5 cannot serve both Penzance and Aberdeen. The refinement is to fit the amplitude to local daylight data.
Model-criticism marks here work as they do everywhere else. Name the assumption, say where it fails, propose a bounded fix.
ASSESSMENT FOCUS
- Read a model centre line first, then amplitude, then period. Each is worth a mark and none of them ever moves house within the formula.
- The period is 2π over the coefficient of t, in radians. Check the mode before you touch the calculator.
- Solve height and depth equations with the widen, substitute and translate routine, keeping exact multiples of π for as long as you can.
- “First time” means the smallest positive solution. If asked for the later ones, give them with the period attached.
CHECK YOURSELF
A buoy's height above mean sea level is modelled by y = 1.2 sin (0.8t), in metres and seconds. Find the amplitude, the period, and the first time the buoy is 0.6 m above mean level.
Show a hint
sin = ½ has a familiar exact angle.
Show the answer
Amplitude 1.2 m, and the period is 2π/0.8 = 7.85 s.
For 0.6 m you need sin (0.8t) = ½, so 0.8t = π/6 and t = 0.654 s.
The next crossing follows at 0.8t = 5π/6. A periodic model always carries a whole family of solutions behind the first one.
Centre line, amplitude, period: three dials, three separate homes in the formula.
Fold two-term models with the harmonic form, then read the extremes straight off R.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the trigonometric modelling questions page.
CHECK YOUR PROGRESS
Rate how confident you feel with each objective for this lesson. Ratings are saved in this browser, on this device, unless you sign in.
- Build and read wheel-style models of the form k − R cos ωt, in radians.
- Extract centre, amplitude, period and phase from any periodic model.
- Use the harmonic form to analyse a model carrying two trig terms.
Open the full revision checklist to see every objective in the course in one place.