Using Maclaurin's series, find the first 4 (non zero) terms...

Using Maclaurin's series, find the first 4 (non zero) terms for the function $f(x)=\sin x$ $$ \begin{array}{cc} f(x)=\sin x & f(0)=\sin 0=0 \\ f^{\prime \prime}(x)=-\sin x \quad f^{\prime \prime}(0)=-\sin 0=0 \\ f^{\prime \prime \prime}(x)=-\cos x \quad f^{\prime \prime \prime}(0)=-\cos 0=-1 \\ f^{\mathrm{iv}}(x)=\sin x \quad & f^{\mathrm{i}}(0)=\sin 0=0 \\ f^{\mathrm{v}}(x)=\cos x & f^{\mathrm{v}}(0)=\cos 0=1 \\ f^{\mathrm{vi}}(x)=-\sin x & f^{\mathrm{vi}}(0)=-\sin 0=0 \\ f^{\mathrm{vii}}(x)=-\cos x & f^{\mathrm{vii}}(0)=-\cos 0=-1 \end{array} $$ Substituting the above values into Maclaurin's series of equation (5) gives: $$ \begin{aligned} \sin x=0+x(1)+\frac{x^{2}}{2 !}(0)+\frac{x^{3}}{3 !}(-1)+\frac{x^{4}}{4 !}(0) \\ +\frac{x^{5}}{5 !}(1)+\frac{x^{6}}{6 !}(0)+\frac{x^{7}}{7 !}(-1)+\cdots \\ \text { i.e. } \sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\cdots \end{aligned} $$

Using Maclaurin's series, find the first 4 (non zero) terms for the function $f(x)=\sin x$
$$
\begin{array}{cc}
f(x)=\sin x & f(0)=\sin 0=0 \\
f^{\prime \prime}(x)=-\sin x \quad f^{\prime \prime}(0)=-\sin 0=0 \\
f^{\prime \prime \prime}(x)=-\cos x \quad f^{\prime \prime \prime}(0)=-\cos 0=-1 \\
f^{\mathrm{iv}}(x)=\sin x \quad & f^{\mathrm{i}}(0)=\sin 0=0 \\
f^{\mathrm{v}}(x)=\cos x & f^{\mathrm{v}}(0)=\cos 0=1 \\
f^{\mathrm{vi}}(x)=-\sin x & f^{\mathrm{vi}}(0)=-\sin 0=0 \\
f^{\mathrm{vii}}(x)=-\cos x & f^{\mathrm{vii}}(0)=-\cos 0=-1
\end{array}
$$
Substituting the above values into Maclaurin's series of equation (5) gives:
$$
\begin{aligned}
\sin x=0+x(1)+\frac{x^{2}}{2 !}(0)+\frac{x^{3}}{3 !}(-1)+\frac{x^{4}}{4 !}(0) \\
+\frac{x^{5}}{5 !}(1)+\frac{x^{6}}{6 !}(0)+\frac{x^{7}}{7 !}(-1)+\cdots \\
\text { i.e. } \sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\cdots
\end{aligned}
$$

Key Concepts

Pattern of Derivatives in Trigonometric Functions

Trigonometric functions like sine exhibit cyclic derivative patterns, meaning their derivatives repeat in a predictable manner. This regularity simplifies the series expansion of trigonometric functions since many of the derivatives are zero or follow a consistent sign pattern, allowing for an efficient construction of the series with non-zero terms.

Factorial in Series Expansion

Factorials appear in the denominator of each term in a Maclaurin series, normalizing the terms. The factorial n! reflects the combinatorial aspect of how the function's changes accumulate in higher order derivatives and ensures the series converges appropriately for many functions.

Maclaurin Series

The Maclaurin series is a specific type of Taylor series centered at zero. It represents a function as an infinite sum of terms derived from the function's derivatives evaluated at zero, multiplied by powers of the variable. This series provides a powerful tool to approximate functions and study their behavior in the vicinity of the origin.

Derivative Evaluation at a Point

Evaluating the derivatives of a function at a specific point, especially at zero for a Maclaurin series, is essential because each term in the series is constructed from these derivatives. This process simplifies the task of building the series expansion by systematically using the known values of the function's derivatives at that point.

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