Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Identify harmonic numbers and amplitudes.
- Evaluate a signed sum.
- Separate a bound from an attained maximum.
Before you start
Sine at quadrantal angles, fractions, and sinusoidal periods.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For F(t) = sin t + (1/3)sin3t + (1/5)sin5t, with phases in radians and t in seconds, find the component periods, common repetition period, and F(π/2).
Why this math matters
A complex periodic shape can be assembled from simpler oscillations. A finite Fourier model makes this idea visible through arithmetic without assuming that every repeating signal is a single sine wave.
Set up the model
A useful answer starts with clear assumptions:
- The fundamental angular frequency is 1 rad/s.
- The three terms are added linearly with the written coefficients.
- This finite model is not asserted to equal any measured sound or ideal square wave.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Build a new waveform from three sine waves
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For F(t) = sin t + (1/3)sin3t + (1/5)sin5t, with phases in radians and t in seconds, find the component periods, common repetition period, and F(π/2).
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Read the components
T₁ = 2π s; T₃ = 2π/3 s; T₅ = 2π/5 s
The third and fifth harmonics oscillate faster, while their amplitudes are smaller: 1/3 and 1/5.
Evaluate one instant
F(π/2) = 1 − 1/3 + 1/5 = 13/15
The component phases are π/2, 3π/2, and 5π/2. Their sine values are 1, −1, and 1, so cancellation matters.
Describe repetition and a bound
F(t + 2π) = F(t); |F(t)| ≤ 1 + 1/3 + 1/5 = 23/15
All components repeat after 2π seconds. Adding their absolute amplitudes gives an upper bound, not proof that all peaks coincide.
The result
The fundamental repetition period is 2π seconds; F(π/2) = 13/15 and |F(t)| never exceeds 23/15.
The combined curve changes shape because component peaks occur at different times. A finite harmonic sum remains continuous, so it cannot exactly reproduce a discontinuous square wave.
Common mistakes to catch
- Adding amplitudes without evaluating phase misses negative components.
- A theoretical upper bound need not be an attained maximum.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find F(π) for this model.
Show a hint
Evaluate sine at π, 3π, and 5π.
Reveal answer and explanation
0
All three sine values vanish.
Practice 2
If the third term is omitted and the second is subtracted, find sin(π/2) − sin(3π/2)/3.
Show a hint
The second sine value is −1.
Reveal answer and explanation
4/3
1 − (−1/3) = 4/3, showing how a coefficient sign changes the combined wave.
Take the idea with you
Plot component values and their sum at the same times before interpreting a harmonic model.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
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