A sinusoid fits anything rising and falling by the same amount on a fixed schedule: tides, daylight, temperature, a 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.
Then amplitude
The amplitude is half the gap between them. It is a distance, so never negative.
Then b
Take the period from the data and use b = 360 ÷ period in degrees.
Sine or cosine
Start with cosine if the data begins at a maximum, and sine if it begins at the midline rising. A good choice removes the phase shift.
Starting at a minimum
Data beginning at its lowest point is an upside-down cosine, which is a negative coefficient rather than a shift.
Extract the four parameters from data
Midline from the average of the extremes, amplitude from half their difference, period from the cycle length, shift from where a peak occurs. Four readings and the model is written.
Sine or cosine
Cosine if the data start at a maximum, sine if they start at the midline rising. Choosing the one matching the starting behaviour minimises the shift and the chance of getting it wrong.
What is genuinely sinusoidal
Tides, daylight hours, average monthly temperature, sound tones and alternating current. Anything driven by rotation or steady oscillation tends to be sinusoidal.
Periodic is not the same as sinusoidal
A square wave repeats and is not a sinusoid. Fourier showed such waves are sums of sinusoids, which is the foundation of signal processing, but the single-wave model does not fit them.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Ines Model a Wheel
A wheel carries riders between 2 m and 42 m, once every 8 minutes, starting at the bottom.
- Step 1
The midline is the average: (42 + 2) ÷ 2 = 22.
Step 4: Your Turn
Practice makes it stick.
The Amplitude
Problem 1 of 2
Max 42 and min 2. What is the amplitude?
The Value of b
Problem 2 of 2
A cycle takes 8 minutes. What is b, in degrees per minute?
Build the Model
1 of 8
Max 20, min 4. Amplitude?
2 of 8
Max 20, min 4. Midline?
3 of 8
A cycle takes 12 hours. What is b, in degrees per hour?
4 of 8
h = 5 sin(60t) + 9. Maximum value?
5 of 8
h = 5 sin(60t) + 9. Period in units of t?
6 of 8
h = −20 cos(45t) + 22 at t = 0. What is h?
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
A wheel runs between 1 m and 31 m. Midline in metres?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Max 30 and min 10. What is the amplitude?
Question 2 of 2
Data starts at its minimum. What 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.