Fit a periodic model and test its assumptions

Try

A fictional rotating display has a marker whose height varies between 1 m and 9 m. It reaches its maximum at time zero and repeats every 12 seconds. Find a cosine model before calculating any intermediate height. The midline is 5 m and amplitude 4 m, so H(t)=5+4cos(πt/6) fits these assumptions.

Predict

Predict the height after a quarter-period and a half-period. A quarter-period is 3 seconds, giving midline height 5. A half-period is 6 seconds, giving minimum height 1. These landmarks can detect a mistaken angular frequency without plotting a dense curve.

Experiment

Sketch a physical circle with centre at height 5 and radius 4, then sketch the height against time separately.

Time in seconds	0	3	6	9	12
Height in metres	9	5	1	5	9

Now change the starting condition to “the marker begins at the bottom.” Use H(t)=5−4cos(πt/6) and recompute the table. The range and period stay the same, while the phase changes. This is a different trajectory through the same possible heights.

Understand

From a maximum M and minimum m, a symmetric sinusoidal model has midline (M+m)/2 and amplitude (M−m)/2. If the period is T>0, angular frequency is 2π/T in radians per time unit. A cosine model with a maximum at time h is k+Acos(2π(t−h)/T) for A>0.

These parameters describe the assumed family. They do not prove that measured behavior is sinusoidal. Real periodic signals can have sharp peaks, unequal rise and fall durations or changing periods. A repeating pattern is not necessarily a sine wave.

Sparse observations also create ambiguity. Measurements 3, 7 and 3 at times 0, 2 and 4 fit 5−2cos(πt/2), but they also fit 5−2cos(3πt/2). The second model oscillates more rapidly between the sampled times. Additional observations between those times can distinguish them. This sampling ambiguity is a reason to inspect the process and sampling schedule, not just fit a smooth curve.

For a wheel model, height is a projection of circular motion. Constant angular speed produces a sinusoidal height, but changing speed generally does not produce a sine of a linear time expression. The radius, centre height, initial angle and time unit must be stated. A physical height cannot become negative if the model claims a wheel remains above ground.

Apply

The practice asks you to construct a wheel model and challenge a three-point fit. Give units and assumptions with predictions. Compare at least one maximum, one minimum and one intermediate value against the context. A formula that matches a peak but fails the starting direction is not a complete model.

Continue

Triangle measurement uses the same trigonometric functions with geometric constraints. In the sine law, an inverse function’s second possible angle can represent a second real triangle, so branch checking becomes a physical feasibility problem.