Taylor series for sin(x) and cos(x), Single Variable Calculus

Let's compute the Taylor series (or Maclaurin series) for f(x)=sin(x) and g(x)=cos(x) centered at x=0. We compute the Maclaurin series for sin(x) using the definition of Taylor coefficients: we start with the function 𝑓(𝑥)=sin(𝑥) and its derivatives, evaluating them at 𝑥=0, until we observe a cyclic pattern in the derivatives. We then use the ratio test to show thhat the radius of convergence is infinite, so we have convergence for all x in (-∞,∞).

For our second function, instead of starting from scratch, we utilize the fact that cos(𝑥) is the derivative of sin(𝑥). By differentiating the Taylor series of sin(𝑥), we obtain the series for cos(𝑥), inheriting the same radius of convergence as for 𝑓(𝑥)=sin(𝑥)

We will see visualizations that show that we need more and more terms to see convergence away from x=0. We discuss the odd symmetry of sin(𝑥) and the even symmetry of cos(𝑥), highlighting how these symmetries are reflected in their respective Taylor series.

We then do another example, this time sin(x) centered at x=π/2.

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