Forman Acton’s book *Numerical Methods that Work* describes Chebyschev polynomials as

cosine curves with a somewhat disturbed horizontal scale, but the vertical scale has not been touched.

The relation between Chebyshev polynomials and cosines is

*T _{n}*(cos θ) = cos(

*n*θ).

Some sources take this as the definition of Chebyshev polynomials. Other sources define the polynomials differently and prove this equation as a theorem.

It follows that if we let *x* = cos θ then

*T _{n}*(

*x*) = cos(

*n*arccos

*x*).

Now sin *x* = cos(π/2 − *x*) and for small *x*, sin *x* ≈ *x*. This means

arccos(*x*) ≈ π/2 − *x*

for *x* near 0, and so we should expect the approximation

*T _{n}*(

*x*) ≈ cos(

*n*(π/2 −

*x*)).

to be accurate near the middle of the interval [−1, 1] though not at the ends. A couple plots show that this is the case.