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Form and use periodic sine and cosine models

Start a set of 8
Step 3 of 5

Find the amplitude from a maximum of 30 and a minimum of 10

\[A=\frac{30-10}{2}=?\]
Year 12, VCE Units 3 & 4, Chapter 2, 4 difficulty levels, Calculator allowedBuild a worksheet

A periodic sine or cosine model

A periodic model describes a bounded pattern that repeats about a stable mean. Find the mean and amplitude from the extrema, use a full cycle for the period, then align sine or cosine with the phase anchor

\[y(t)=aF\left(n\pi t+h\right)+k\]
  1. Use repeated bounded data A stable rise and fall that repeats supports a periodic model \(\displaystyle 12,16,20,16,12,16,20,16\)
  2. Find the mean The mean is halfway between the maximum and minimum \(\displaystyle k=\frac{M+m}{2}=\frac{30+10}{2}=20\)
  3. Find the amplitude The positive amplitude is half the distance between the extrema \(\displaystyle A=\frac{M-m}{2}=\frac{30-10}{2}=10\)
  4. Use a full cycle The period is the time for one complete repeat, from one matching phase point to the next \(\displaystyle T=12\text{ h},\quad n=\frac{2}{T}=\frac{1}{6}\)

Worked example

A sensor displacement has maximum 14 mm, minimum −2 mm and period 8 s. It crosses its mean value while increasing one second after the start. Form the canonical sine model

\[M=14,\quad m=-2,\quad T=8\text{ s},\quad t=1\text{ s}\]
  1. \(\displaystyle A=\frac{14-(-2)}{2}=8\)The amplitude is half the distance between the maximum and minimum
  2. \(\displaystyle k=\frac{14+(-2)}{2}=6\)The mean is the midpoint of the two extrema
  3. \(\displaystyle T=8\Rightarrow n=\frac{2}{8}=\frac{1}{4}\)The coefficient n is found from the full period in the canonical form
  4. \(\displaystyle \frac{h}{\pi}=-n(1)=-\frac{1}{4}\equiv\frac{7}{4}\pmod{2},\quad y(t)=8\sin\left(\frac{\pi}{4}t+\frac{7\pi}{4}\right)+6\)A rising mean crossing occurs when the sine input is 0 and increasing, and the phase is written in one period

Common mistakes

Calling one turn periodic
✕\(\displaystyle 4,7,10,7,4\Rightarrow\text{periodic}\)✓\(\displaystyle 4,7,10,7,4,7,10,7\Rightarrow\text{periodic pattern supported}\)
One rise and fall does not establish a repeating cycle or stable mean
Using an extremum as the mean
✕\(\displaystyle M=30,\quad m=10\Rightarrow k=30\)✓\(\displaystyle k=\frac{M+m}{2}=\frac{30+10}{2}=20\)
The equilibrium or mean is the midpoint of the maximum and minimum, not an extremum
Using peak-to-trough as the period
✕\(\displaystyle T=\frac{12}{2}=6\)✓\(\displaystyle T=12\)
The period covers a complete repeat, while the time from a maximum to a minimum is half a period

Difficulty levels

introIdentify repeated bounded data and interpret its amplitude, mean and period
standardForm a canonical sine or cosine rule from extrema, period and a phase anchor, then evaluate at an exact time
challengeFind an extremum time, level crossing or threshold duration in a stated modelling domain
examChoose the model and rule, then calculate one numerical consequence

Questions about this topic

How is the mean value found?+

The mean is halfway between the maximum and minimum

What does amplitude measure?+

Amplitude is the distance from the mean to an extremum, so it is half the difference between the maximum and minimum

Why is the period a full cycle?+

The period is the time for the same phase to repeat. The time from a maximum to a minimum is only half a cycle

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 2: Functions and graphs

In this chapter

Read restricted graphs and horizontal level setsFind the gradient of a line through two pointsFind the equation of a line through two pointsSolve a pair of simultaneous linear equationsRead the vertical-axis intercept and gradient from an equationRead the point of intersection of two linesConstruct a linear model from a descriptionInterpret the graph of a linear modelTransform a quadratic and find its key featuresDetermine a quadratic rule from exact informationAnalyse signs and solve a quadratic inequalityAnalyse a cubic graph or determine its ruleInterpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityApproximate solutions from intersections of two graphsClassify a relation and state its domain and rangeEvaluate and determine function rulesApply and describe transformations of a functionAnalyse the general rational-power familyFind the maximal domain of an algebraic functionAnalyse reciprocal and square-reciprocal functionsAnalyse sideways-parabola and root-function familiesDetermine a power-family rule from features and pointsWork with sets and interval notationSimplify, compare and operate with surdsCombine function rules and ordinatesRecognise equivalent polynomial formsUse a piecewise linear modelEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsTransform and reconstruct sine and cosine functionsFeatures and transformations of tangent functionsForm and use periodic sine and cosine modelsAnalyse exponential and logarithmic graphsInterpret and use an exponential modelSelect and interpret a function modelEvaluate and interpret a restricted rational model