(12C) SIN COS TAN

[Albert Chan]

sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ... = x * (1 - x²/6 / (1 + x²/20)) + O(x^7)

\(\displaystyle s(x)
= x \left( 1-\frac{20}{6}\left(\frac{\frac{x^2}{20}}{1+\frac{x^2}{20}} \right) \right)
= \frac{x}{3} \left( 3-10\left(1 - \frac{1}{1+\frac{x^2}{20}} \right) \right)
= \frac{x}{3} \left(-7 + \frac{10}{1+\frac{x^{2}}{20}}\right)
\)


cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ... = 1 - x²/2 * (1 - x²/12 / (1 + x²/30)) + O(x^8)

\(\displaystyle c(x)
= 1 - \frac{x^2}{2} \left(1 - \frac{30}{12} \left(\frac{\frac{x^2}{30}}{1+\frac{x^2}{30}} \right)\right)
= 1 - \frac{x^2}{4} \left(2 - 5 \left(1 - \frac{1}{1+\frac{x^2}{30}} \right)\right)
= 1 - \frac{x^2}{4} \left(-3 + \frac{5}{1+\frac{x^2}{30}} \right)
\)

We may also reuse code: versin(x) = 1 - cos(x) = 2*sin(x/2)^2

\(\cos(x) ≈ 1 - 2\, s(\frac{x}{2})^2 \)