Quick Chebyshev value

Second-kind Chebyshev polynomials often surface in HSC Extension 1 and 2 work when exploring trigonometric identities or series expansions. This short exercise gives you a chance to practise direct algebraic substitution with fractions—an essential skill that sets you up for success when the algebra gets heavier. This type of question often appears in the Extension 1 topic Further Trigonometry, where Chebyshev polynomials are used to express $\cos n\theta$ as a polynomial in $\cos\theta$.

Problem Statement

For the second-kind Chebyshev polynomial

$$U_2(x)=4x^2-1,$$

find $U_2\left(\frac12\right)$.

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Hints

Substitute $x=\frac12$ carefully.

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Solutions

We substitute $x = \frac12$ directly into the expression. Since $\left(\frac12\right)^2 = \frac14$, the $4x^2$ term becomes $4 \times \frac14 = 1$, cancelling neatly with the $-1$ to give zero.

$$U_2\left(\frac12\right)=4\left(\frac14\right)-1=1-1=0.$$

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Takeaways

• For a given Chebyshev polynomial, direct substitution is often enough.
• Square fractions before multiplying by the coefficient.

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Further Readings

HSC Trigonometry, HSC Last Resorts, HSC Mechanics, HSC Vectors