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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Derive Chebyshev polynomials from a three-term recurrence.
- Justify the conclusion "T₂=2x²−1 and T₃=4x³−3x" using the stated assumptions.
Before you start
Cosine addition formulas and polynomials.
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
Use Tₙ(cos θ)=cos(nθ) to find T₂ and T₃.
Why this math matters
Derive Chebyshev polynomials from a three-term recurrence. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- n is a nonnegative integer.
- The cosine interpretation initially uses x∈[−1,1]; the polynomial identity extends algebraically.
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.
Generate polynomial identities from cosine addition
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Use Tₙ(cos θ)=cos(nθ) to find T₂ and T₃.
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.
Build the model
cos((n+1)θ)+cos((n−1)θ)=2cos θ cos(nθ)
Adding two angle-sum identities cancels the sine terms.
Work through the mathematics
Tₙ₊₁(x)=2xTₙ(x)−Tₙ₋₁(x), with T₀=1,T₁=x
The identity becomes a polynomial recurrence.
Check the conclusion
T₂=2x²−1 and T₃=4x³−3x
Successive substitution recovers familiar double- and triple-angle formulas.
The result
T₂=2x²−1 and T₃=4x³−3x
Successive substitution recovers familiar double- and triple-angle formulas.
Common mistakes to catch
- A recurrence needs its starting values.
- The boundedness claim does not extend unchanged outside [−1,1].
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
Compute T₄.
Show a hint
Use 2xT₃−T₂.
Reveal answer and explanation
8x⁴−8x²+1
Expanding 2x(4x³−3x)−(2x²−1) gives the result.
Practice 2
Why is |Tₙ(x)|≤1 on [−1,1]?
Show a hint
Write x=cos θ.
Reveal answer and explanation
Because |cos(nθ)|≤1
Every x in that interval has an angular representation.
Take the idea with you
Use recurrence evaluation to avoid explicitly computing a high multiple angle.
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.
Up next: Correct double counting in event unions
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