A sinusoid fits anything that rises and falls by the same amount on a fixed schedule: tides, daylight hours, temperature, a Ferris wheel.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Midline first
The midline is the average of the maximum and minimum, d = (max + min) ÷ 2.
Then amplitude
The amplitude is half the gap, a = (max − min) ÷ 2. It is a distance, so it is never negative.
Then b
Take the period from the data and use b = 360° ÷ period.
Sine or cosine
Start with cosine if the data begins at a maximum, and with sine if it begins at the midline going up. Choosing well means no phase shift at all.
Fit the four parameters
Midline is the average of maximum and minimum. Amplitude is half their difference. Period comes from how long a full cycle takes. Phase shift comes from where a peak or zero occurs. Four readings give the model.
Which function to start from
Use cosine if the data starts at a maximum and sine if it starts at the midline rising. Choosing the function that matches the starting behaviour minimises the phase shift and the chance of getting it wrong.
What is genuinely sinusoidal
Tides, daylight hours, average monthly temperature, alternating current and sound tones. Anything driven by rotation or oscillation tends to be sinusoidal, which is why the model is so widely applicable.
And what is not
A pattern that repeats but with sharp corners — a square wave, or a sawtooth — is periodic but not sinusoidal. Fourier showed such waves are sums of sinusoids, which is the foundation of all signal processing.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Tomas Model a Ferris Wheel
A wheel carries riders between 2 m and 42 m above the ground, once every 8 minutes, starting at the bottom.
- Step 1
The midline is the average of the two heights: (42 + 2) ÷ 2 = 22.
Step 4: Your Turn
Practice makes it stick.
The Tide
Problem 1 of 2
High tide is 6 m and low tide is 2 m. What is the amplitude?
m
The Daylight
Problem 2 of 2
Daylight runs between 9 and 15 hours over a year. What is the midline, in hours?
hours
Build the Model
1 of 8
Max 20, min 4. Amplitude?
2 of 8
Max 20, min 4. Midline?
3 of 8
The cycle takes 12 hours. What is b, in degrees per hour?
4 of 8
h = 5 sin(60t) + 9. What is the maximum?
5 of 8
h = 5 sin(60t) + 9. What is the period, in units of t?
6 of 8
A wheel runs between 1 m and 31 m. What is the midline, in metres?
7 of 8
Put the modelling steps in order.
- 1Find the amplitude from half the gap.
- 2Find b from 360° divided by the period.
- 3Choose sine or cosine from where the data starts.
- 4Find the midline from the average of max and min.
8 of 8
h = −20 cos(45t) + 22. What is the height at t = 0, in metres?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Max 30, min 10. What is the amplitude?
Question 2 of 2
Data starts at its maximum. Which model avoids a phase shift?
What You Learned
- Midline is the average of max and min; amplitude is half their gap.
- b is 360° divided by the period.
- Choosing the right starting curve removes the need for a phase shift.