**Problem:**Compute the derivative of $\sin x$ in two ways:(i) Using the chain rule.(ii) Taking the derivative of the series for $\sin x$ and comparing it to the series for $\cos x$.Express your answer in terms of the $\cos$ function. ### Taylor Series for Sine and Cosine Functions#### Sine Function (sin x)The Taylor series expansion for the sine function is given by:\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]This can be expressed in summation notation as:\[ \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n \cdot x^{2n+1}}{(2n+1)!} \]#### Cosine Function (cos x)The Taylor series expansion for the cosine function is given by:\[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \]This can be expressed in summation notation as:\[ \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n \cdot x^{2n}}{(2n)!} \]### ExplanationThese series are infinite sums used to approximate the values of the trigonometric functions sine and cosine. Each term in the series is derived from the derivatives of the sine and cosine functions evaluated at zero, and factorials in the denominators provide normalization. The alternating signs are due to the negative derivatives at each step.Taylor series are crucial in mathematics and physics for approximating function values when direct calculation is difficult.

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Compute the derivative of Sin x in two ways: (i) Using the chain rule. (ii) Taking the derivative of the series for Sin x and comparing it to the series for Cos x. Express your answer in terms of the Cos function.

Compute the derivative of Sin x in two ways: (i) Using the chain rule. (ii) Taking the derivative of the series for Sin x and comparing it to the series for Cos x. Express your answer in terms of the Cos function.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Problem:**

Compute the derivative of $\sin x$ in two ways:

(i) Using the chain rule.

(ii) Taking the derivative of the series for $\sin x$ and comparing it to the series for $\cos x$.

Express your answer in terms of the $\cos$ function.

### Taylor Series for Sine and Cosine Functions

#### Sine Function (sin x)

The Taylor series expansion for the sine function is given by:

\[
\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots
\]

This can be expressed in summation notation as:

\[
\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n \cdot x^{2n+1}}{(2n+1)!}
\]

#### Cosine Function (cos x)

The Taylor series expansion for the cosine function is given by:

\[
\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots
\]

This can be expressed in summation notation as:

\[
\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n \cdot x^{2n}}{(2n)!}
\]

### Explanation

These series are infinite sums used to approximate the values of the trigonometric functions sine and cosine. Each term in the series is derived from the derivatives of the sine and cosine functions evaluated at zero, and factorials in the denominators provide normalization. The alternating signs are due to the negative derivatives at each step.

Taylor series are crucial in mathematics and physics for approximating function values when direct calculation is difficult.

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