Advanced Chebyshev Polynomial Calculator

Calculator Inputs

The page uses a stacked section flow, while the calculator fields expand to three columns on large screens, two on tablets, and one on mobile.

Polynomial Kind

Degree n

Input Mode

x Value

Angle θ

Angle Unit

Graph Start

Graph End

Graph Sample Points

Example Data Table

Polynomial	x	Expanded Form	Value
T3(x)	0.5	4x^3 - 3x	-1
T4(x)	0.5	8x^4 - 8x^2 + 1	-0.5
U3(x)	0.25	8x^3 - 4x	-0.875
U4(x)	0.5	16x^4 - 12x^2 + 1	-1

Formula Used

First kind recurrence:

T0(x) = 1, T1(x) = x

Tn(x) = 2xTn-1(x) - Tn-2(x)

Second kind recurrence:

U0(x) = 1, U1(x) = 2x

Un(x) = 2xUn-1(x) - Un-2(x)

Trigonometric identities:

Tn(cos θ) = cos(nθ)

Un(cos θ) = sin((n + 1)θ) / sin(θ)

Derivative idea:

The derivative is formed from the expanded polynomial coefficients.

For the first kind, dTn/dx also equals nUn-1(x).

How to Use This Calculator

Select the polynomial family: first kind Tn or second kind Un.
Enter the degree n. Values from 0 through 20 are allowed.
Choose whether to enter x directly or derive x from an angle θ.
Provide the graph interval and the number of sample points.
Press Calculate to display the result block above the form.
Review the polynomial expansion, roots, derivative, recurrence table, and graph.
Use the CSV button for data export and the PDF button for a report.

FAQs

1. What does this calculator compute?

It evaluates Chebyshev polynomials of the first or second kind, expands coefficients, computes the derivative, lists theoretical roots, estimates critical points, and draws the curve.

2. What is the difference between Tn and Un?

Tn polynomials satisfy Tn(cos θ) = cos(nθ). Un polynomials satisfy Un(cos θ) = sin((n+1)θ)/sin(θ). They share a similar recurrence but start from different initial terms.

3. Why can I enter an angle instead of x?

Chebyshev polynomials are often expressed with x = cos(θ). Angle mode lets you work naturally with the trigonometric identity and then evaluates the polynomial using the derived x value.

4. Are the listed roots exact?

The roots are generated directly from the known closed-form formulas for Chebyshev polynomials. They are displayed numerically, so the shown decimals are approximations of exact trigonometric values.

5. How are critical points found?

The derivative polynomial is formed from the expanded coefficients. The calculator then searches numerically inside [-1, 1] to locate derivative zeros and reports the corresponding function values.

6. Why limit the degree to 20?

Higher degrees can produce large coefficients, longer tables, and heavier plotting. A cap of 20 keeps the page responsive while still covering many practical approximation and numerical analysis tasks.

7. What does the derivative output mean?

The derivative output shows the symbolic derivative polynomial and its value at your selected x. This helps measure slope, turning behavior, and local sensitivity near the evaluation point.

8. What is included in the CSV and PDF exports?

The exports include the summary metrics, coefficient table, recurrence values, critical points, roots, and graph sample data so you can save or reuse the results elsewhere.