Find every threshold time in a stated interval

Try

A synthetic depth model is h(t)=5+3sin(π(t−1)/4), with depth in meters and time in hours. Over the observation window 0≤t≤10, when does the model reach 6.2 meters, and when is it at least that deep? These are different questions: the first asks for isolated crossing times, the second for intervals.

The model has midline 5 meters, amplitude 3 meters and period 8 hours. The input angle is in radians; the factor π/4 has units radians per hour. The shifted rising midline crossing occurs at t=1. The example is a controlled mathematical model, not a calibrated tide forecast.

Predict

The range is [2,8], so 6.2 is reachable. Normalize the target before using a calculator: subtract 5 and divide by 3 to get sin θ=0.4, where θ=π(t−1)/4. Predict two crossings per full cycle. Because the window is longer than one cycle but starts before the shifted cycle origin, determine which parts of neighboring cycles it includes rather than multiplying a guessed count.

Experiment

Set a calculator to radian mode. Let α=arcsin(0.4)≈0.41151685, keeping extra digits internally. The two angle families give two time families:

t=1+4α/π+8k, and t=5−4α/π+8k,

where k is an integer. Put a=4α/π≈0.52395952. Inspect the candidates before rounding.

Cycle shift k	First family 1+a+8k	Second family 5−a+8k
−1	About −6.476	About −3.524
0	About 1.524	About 4.476
1	About 9.524	About 12.476

The three admissible crossing times in [0,10] are approximately 1.52, 4.48 and 9.52 hours. Substitute an unrounded candidate into the original model to check the normalization and phase. A degree-mode inverse value must first be converted or used with a consistently degree-based formula; inserting degrees into this radian formula gives an incorrect time.

Understand

Completeness comes from circle geometry. A horizontal line at sine level 0.4 meets the circle at α and π−α in each revolution. Adding all integer full turns therefore gives every angle solution. The transformed time window is −π/4≤θ≤9π/4. Filtering the two families against either this angle window or the original time window yields the same three candidates. Rounding before filtering can incorrectly include or exclude a near-boundary time, so keep exact expressions until the admissible list is known.

For an inequality, sine is at least 0.4 on the arcs from α to π−α, repeated every 2π. Translating those arcs into time and intersecting with [0,10] gives [1+a,5−a] and [9+a,10]. Thus the qualifying intervals are approximately [1.52,4.48] and [9.52,10] hours. The last interval ends at the observation boundary, not at a falling crossing. The symbol ≥ includes equality endpoints; a strict threshold would exclude crossing times but still include t=10 if its value is strictly above the threshold.

Check interval interiors as an error diagnosis. At t=3 the sine angle is π/2 and the depth is 8, so the first interval should qualify. At t=7 the depth is 2 and should not qualify. At t=10 the model is above 6.2, confirming the final clipped interval. This does not replace the all-solution proof, but catches branch and sign mistakes.

If observations occur only at integer hours, the qualifying sample times are 2,3,4 and 10. These samples are not the continuous qualifying intervals. Likewise, a model may predict hundredths of an hour while measured parameters support much less precision. Requested numerical rounding is a reporting convention, not evidence of instrument accuracy or perfect periodicity.

Apply

Repeat the full process for a target above 8 meters. Explain why a calculator domain error expresses an impossible target under the model rather than a missing solution branch. Then use the practice temperature model to handle cosine, including an interval that already qualifies at t=0. An inverse key gives one angle; symmetry, repetition and the time domain supply the decision.

Continue

Attempt the mixed assessment with calculator mode, exact expressions, complete families and units recorded. The four lessons address selected school trigonometric-function objectives; they do not establish full school or AP readiness. Independent performance and realistic model validation need additional evidence beyond these public worked cases.