A synthetic sensor is sampled at inputs x=0,1,2,3,4,5,6,7,8. Its responses are 13.2,9.9,7.1,9.8,13.0,10.2,6.9,10.1,12.8 sensor units. The high readings near 0,4,8 suggest a period near four input units, but the readings are not exact repetitions. First sketch the raw points. Do not draw a smooth curve and then treat the curve as an additional observation.
The task is to compare specified periodic candidates, not to announce a period from visual resemblance alone. A longer-period candidate may fit a broad trend; a different shorter period may agree at every sampled input. We will retain equations and residuals and then predict at a new input that was not used for fitting.
Predict
Predict which candidate period, 4 or 8 input units, will have smaller training error. Then separately predict whether that candidate will give the better response at x=0.5. The first question concerns fitting the observed grid; the second concerns a new input. They can have different answers.
For each supplied period P, the workbench fits y=c+a cos(2πx/P)+b sin(2πx/P). The period is fixed while c,a,b are chosen to minimize the sum of squared original-response residuals. Here x and P have the same input units, so 2πx/P is an angle in radians. The coefficients c,a,b have response units. This is least squares conditional on a chosen period, not simultaneous global optimization of every sinusoidal parameter.
Experiment
Keep the supplied rows and candidate periods 4 and 8. Select Reveal both periodic fits, then record each equation, residual list and SSE. Inspect the two candidate curves on common axes and the numerical table. Use New x to enter 0.5 and select Predict with both fixed models. Do not append a validation response to the fitting rows.
Compare two supplied periods
Predict which candidate period will describe the observations more convincingly before revealing calculations. The default pairs are noisy synthetic data. State any contextual units and the observation process in your written response; the tool supplies no physical interpretation.
Use 6–40 pairs with both coordinates between −100 and 100. Repeated rows remain observations, not automatically independent measurements. The two candidate periods are your inputs, not periods discovered by a global search.
If you select a period using a validation observation, that observation has become model-selection evidence. A later independent evaluation requires fresh evidence. Record raw rows, supplied periods, equations, residuals and exposure to checks in your written response. Editing data or periods clears all prior results; query edits clear only query results. Reset restores inputs without calculating. This temporary workbench does not save responses or award mastery.
For P=4, the fitted rule is y=10+3cos(πx/2)+0.05sin(πx/2), with SSE 0.19 squared sensor units. Its prediction at 0.5 is 10+3.05/√2≈12.15667568. For P=8, the coefficients are approximately c=10.27272727, a=0.54545455, b=−0.05606602, with SSE 42.73288095. That candidate predicts approximately 10.75520603 at 0.5.
Candidate period
Training SSE
Prediction at unused 0.5
4
0.19
Approximately 12.15668
8
Approximately 42.73288
Approximately 10.75521
Now compare the supplied new measurement 12.1±0.2. The allowable interval [11.9,12.3] contains the period-4 prediction and excludes the period-8 prediction. This supports the first candidate among these choices on this check. One measurement is limited evidence, and the synthetic dataset does not represent a validated real sensor system.
Understand
The coefficients can be rewritten as c+Acos(2πx/P−φ), where A=√(a²+b²), A cosφ=a and A sinφ=b. Expanding the shifted cosine verifies the formula. Use both a and b to select the phase quadrant; a simple arctangent ratio loses quadrant information. For the period-4 fit A=√9.0025≈3.00041664 and φ≈0.01666512 radians. Its maximum occurs at input shift Pφ/(2π), repeated every P. Phase angles and time shifts are not interchangeable units.
If a=b=0, the curve is constant: amplitude is zero and phase is undefined. Assigning a phase of zero would invent information. The chosen period is then not identified by the response. If the sampled inputs always represent the same phase, the design may also fail to distinguish coefficients. For example, with P=4, inputs 0,4,8,… all have cosine 1 and sine 0; only c+a is observable. The tool reports unresolved fits rather than manufacturing separate coefficients.
There is a stronger ambiguity even with diverse phases. At every integer n,
cos(3πn/2)=cos(2πn−πn/2)=cos(πn/2),
while sin(3πn/2)=−sin(πn/2). Therefore the period-4 model and the period-4/3 model with the same c,a and opposite b agree at every integer input. No number of additional measurements on that same integer grid can distinguish them if the exact models hold. This is sampling aliasing.
Change the second candidate period to 4/3, entering approximately 1.3333333333333333. The mathematical alias fit has c=10,a=3,b=−0.05 and the same SSE 0.19, but predicts 10−3.05/√2≈7.84332432 at 0.5. The new observation distinguishes these two candidates where the training grid could not. Small floating-point differences from the decimal period are numerical approximations, not physical evidence of a different fit quality.
Apply
Construct a changed exact dataset at x=0,1,2,3,4,5 with response rule y=6+2sin(πx/3). Enter its six values, compare candidate periods 6 and 3, and predict at x=0.5 before computing the exact response from the formula. Keep this formula-derived exercise labeled as synthetic practice, not an independent measurement study.
Check the intended period and new response
The rows are 6,6+√3,6+√3,6,6−√3,6−√3. Use sufficiently precise decimals when entering them. For period 6 the exact coefficients are c=6,a=0,b=2, with zero training SSE; at 0.5 the response is 6+2sin(π/6)=7. The period-3 candidate cannot represent these six observations exactly: inputs 1 and 4 have the same phase for that period but different responses. Its fitted error is therefore positive. A correct training fit still depends on the chosen family, and a finite zero error does not establish an unrestricted physical law.
Retain the scale and sampling plan with a recommendation. A plot window showing many rapid cycles can be misleading at finite resolution; the workbench omits curve strokes when a candidate would exceed its display-resolution bound, keeping the equation and numerical evidence available. Such omission is not evidence that the model has no curve. Do not infer a slowly varying curve from an aliased picture.
Recover features, then challenge an alias
Suppose a sinusoidal model has a maximum of eight at t=1 and the next minimum of two at t=4. The midline is five, the amplitude is three and the three-unit time separation is half a period. A matching model is y=5+3cos(π(t−1)/3), with period six. Substitute both named times to verify the extrema. Assuming these extrema are consecutive is essential: skipping cycles would make the period ambiguous.
A second ambiguity comes from sampling. Compare A(t)=3+4sin(πt/4) and B(t)=3+4sin(5πt/4). At t=2n their phases differ by 2πn, so every even-integer sample agrees. Collecting more samples only on that same grid cannot distinguish these exact candidates. At t=1, however, A predicts 3+2√2 and B predicts 3−2√2. The new sampling location changes what evidence is available.
Move the time control between even and odd inputs. Explain each equality or difference using phases rather than simply reporting the decimals. The control compares stipulated models; it does not fit new observations or identify a universally correct physical frequency. Small floating-point discrepancies at theoretically equal values are not measurement evidence.
The changed practice joins graph recovery to model validation. Keep a record of the features used to construct the equation and the unused input chosen to test alternatives. Once a new measurement guides the choice, it becomes development data. A later independent performance claim needs fresh observations and a stated measurement-error tolerance.
Recover a graph and test ambiguity
Review criterion: Derive midline, amplitude and period from features; use unsampled inputs to test competing models.
Predict, then change one input
A(t)=3+4sin(πt/4)
7
B(t)=3+4sin(5πt/4)
7
On the even-integer grid these exact candidates agree. Tiny decimal differences are roundoff, not evidence against aliasing.
Explain the changed case: A shifted cosine
A sinusoid has consecutive maximum and minimum at (1,8) and (4,2). Which proposed model has the stated features?
Midline (8+2)/2=5
Amplitude (8−2)/2=3
A maximum to the next minimum is half a period.
Write a reason before selecting a claim. Feedback is public practice, not a proof grader.
Inputs and choices are temporary. Nothing is saved, no lesson completion or score is changed, and repeating a public case is not unseen assessment.
Continue
Record which periods were compared, which were not, the coefficient and phase conventions, original-response errors, and every unused prediction before its outcome. If a validation result guides the next period choice, it becomes development evidence; later independent evaluation needs new inputs. Residual plots and a small SSE help inspect a candidate but do not automatically establish a stable periodic process.
Next, construct tangent and cotangent branches. Unlike sine and cosine, those graphs have excluded inputs and unbounded branches. A finite drawing must preserve those breaks rather than connecting every neighboring plotted sample.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Use the periodic fitting workbench with x=0,1,2,3,4,5,6,7,8 and responses 7.1,9.8,10.2,7.2,3.9,4.2,7.1,9.6,10.1 in sensor units. Compare candidate periods 6 and 4 input units. Retain each fitted c+a cos(2πx/P)+b sin(2πx/P), residuals and SSE. Predict at unused x=0.5 before comparing with synthetic reading 8.5±0.2. Tolerance 0.002 for coefficients,SSE and predictions.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A fitted rule is y=4−3 cos(ωx)+4 sin(ωx), with ω=π/3. Express it as 4+Acos(ωx−φ) with A≥0 and φ in [0,2π). State its period and where one maximum occurs.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compare f(x)=2+3 cos(πx/2) and g(x)=2+3 cos(3πx/2). Prove they agree at every integer input, then calculate their predictions at x=0.5. Explain how this affects period selection from integer-spaced observations.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A learner compares only periods 5 and 10, obtaining training SSEs 1.2 and 8.4, then announces the true period is 5. Explain precisely what the calculation supports and design a stronger comparison, including a fresh check and units.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For a fitted model c+a cos(2πx/P)+b sin(2πx/P), explain amplitude and phase when a=b=0. Does the selected P become an identified physical period? Contrast this with a small but nonzero amplitude under measurement noise.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A team samples only at x=0,4,8,12,16,20 and asks for a sinusoidal fit with period 4. Show why the coefficientsc,a,b cannot be separately recovered from those rows. What changed sampling plan could help?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
