Chapter 2: Q2P (page 69)

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

Short Answer

Expert verified

The solution for cos θ and sin θ.

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

Step by step solution

Given Information.

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

Definition of Power series.

A power series is an infinite series which looks like

$\sum_{n = 0}^{\infty} a_{n} {(x - c)}^{n} = a_{0} + a_{1} (x - c) + a_{2} {(x - c)}^{2} + \cdot \cdot \cdot$
where represents the coefficient of the nth term and c is a constant.

Add the given functions.

Write the two equations.

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

Add the functions.

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

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

Subtract the given functions.

Subtract the functions.

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

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

Therefore,

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

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Solve the equations $e^{i θ} = c o s θ + i s i n θ, e^{- i θ} = c o s θ - i s i n θ$ , for cos θ and sin θ 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.

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Solve the equations eiθ=cosθ+isinθ,e-iθ=cosθ-isinθ, for cos θ and sin θ 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

Fluids

Energy Physics

Force

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Study anywhere. Anytime. Across all devices.

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