Modelling with Sinusoids

Fit a wave to something that actually repeats.

12 problems · about 22 minutes · F-TF.B.5, F-IF.B.4

What this lesson teaches

The student builds a sinusoidal model from described periodic data and uses it to predict values.

Straightforward practice. Get the method working first.

5 problems

Max 30, min 10. What is the amplitude?
Answer 10
Why 10.
Data starts at its maximum. Which model avoids a phase shift?
Answer Cosine, because it starts at its peak.
Why Cosine, which is already at its maximum when the input is zero.
Max 20, min 4. Amplitude?
Answer 8
Why (20 − 4) ÷ 2.
Max 20, min 4. Midline?
Answer 12
Why (20 + 4) ÷ 2.
The cycle takes 12 hours. What is b, in degrees per hour?
Answer 30
Why 360 ÷ 12.

The same ideas with more to keep track of.

3 problems

h = 5 sin(60t) + 9. What is the maximum?
Answer 14
Why Midline plus amplitude.
h = 5 sin(60t) + 9. What is the period, in units of t?
Answer 6
Why 360 ÷ 60.
A wheel runs between 1 m and 31 m. What is the midline, in metres?
Answer 16
Why (31 + 1) ÷ 2.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Put the modelling steps in order.
Answer 1. Find the midline from the average of max and min. 2. Find the amplitude from half the gap. 3. Find b from 360° divided by the period. 4. Choose sine or cosine from where the data starts.
Why The starting point only matters once the shape is fixed.
h = −20 cos(45t) + 22. What is the height at t = 0, in metres?
Answer 2
Why cos 0 = 1, so h = −20 + 22.
The Tide: High tide is 6 m and low tide is 2 m. What is the amplitude?
Answer 2 m
Why 2 m.
The Daylight: Daylight runs between 9 and 15 hours over a year. What is the midline, in hours?
Answer 12 hours
Why 12 hours.