Q2P Solve the equations ei胃=cos胃+i... [FREE SOLUTION]

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Chapter 2: Q2P (page 69)

Solve the equations $e^{i 胃} = c o s 胃 + i s i n 胃, e^{- i 胃} = c o s 胃 - i s i n 胃$ , for cos 胃 and sin 胃 and so obtain equations (11.3).

Short Answer

Expert verified

The solution for cos 胃 and sin 胃.

$\cos (胃) = \frac{e^{(胃 i)} + e^{(- 胃 i)}}{2} \sin (胃) = \frac{e^{(胃 i)} - e^{(- 胃 i)}}{2 i}$

Step by step solution

Given Information.

The given equations are $e^{i 胃} = c o s 胃 + i s i n 胃, e^{- i 胃} = c o s 胃 - i s i n 胃$ .

Definition of Power series.

A power series is an infinite series which looks like

${鈭�}_{n = 0}^{鈭�} a_{n} {(x - c)}^{n} = a_{0} + a_{1} (x - c) + a_{2} {(x - c)}^{2} + 路路路$
where represents the coefficient of the nth term and c is a constant.

Add the given functions.

Write the two equations.

$\begin{array}{l} e^{i 胃} = \cos 胃 + i \sin 胃, \\ e^{- i 胃} = \cos 胃 - i \sin 胃 \end{array}$

Add the functions.

$e^{(胃 i)} + e^{(- 胃 i)} = \cos (胃) + i \sin (胃) + \cos (胃) - i \sin (胃) e^{(胃 i)} + e^{(- 胃 i)} = 2 \cos (胃) \cos (胃) = \frac{e^{(胃 i)} + e^{(- 胃 i)}}{2}$

Hence the $\cos (胃)$ is, $\frac{e^{(胃 i)} + e^{(- 胃 i)}}{2}$ .

Subtract the given functions.

Subtract the functions.

$e^{(胃 i)} - e^{(- 胃 i)} = \cos (胃) + i \sin (胃) - \cos (胃) + i \sin (胃) e^{(- 胃 i)} - e^{(- 胃 i)} = 2 i \sin (胃) \sin (胃) = \frac{e^{(胃 i)} - e^{(- 胃 i)}}{2 i}$ \

Hence the $\sin (胃)$ is, $\frac{e^{(胃 i)} - e^{(- 胃 i)}}{2 i}$ .

Therefore,

$\begin{array}{l} \cos (胃) = \frac{e^{(胃 i)} + e^{(- 胃 i)}}{2} \\ \sin (胃) = \frac{e^{(胃 i)} - e^{(- 胃 i)}}{2 i} \end{array}$

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91影视

Solve the equations $e^{i 胃} = c o s 胃 + i s i n 胃, e^{- i 胃} = c o s 胃 - i s i n 胃$ , for cos 胃 and sin 胃 and so obtain equations (11.3).

Short Answer

Step by step solution

Given Information.

Definition of Power series.

Add the given functions.

Subtract the given functions.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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91影视

Solve the equations ei胃=cos胃+isin胃,e-i胃=cos胃-isin胃, for cos 胃 and sin 胃 and so obtain equations (11.3).

Short Answer

Step by step solution

Given Information.

Definition of Power series.

Add the given functions.

Subtract the given functions.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Quantum Physics

Electricity and Magnetism

Famous Physicists

Physical Quantities And Units

Classical Mechanics

Study anywhere. Anytime. Across all devices.

Solve the equations $e^{i 胃} = c o s 胃 + i s i n 胃, e^{- i 胃} = c o s 胃 - i s i n 胃$ , for cos 胃 and sin 胃 and so obtain equations (11.3).