# To prove The formulas cos x = e i x + e − i x 2 and sin x = e i x − e − i x 2 i by the use of Euler formula.

BuyFind

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter I, Problem 48E
To determine

## To prove The formulas cosx=eix+e−ix2 and sinx=eix−e−ix2i by the use of Euler formula.

Expert Solution

### Explanation of Solution

Formula used:

Euler’s formula:

eiy=cosy+isiny

Calculation:

Simplify the expression eix+eix by the use of Euler’s formula as follows.

eix+eix=(cosx+isinx)+(cos(x)+isin(x))=cosx+isinx+cosxisinx[cos(x)=cosxandsin(x)=sinx]=cosx+cosx=2cosx

Thus, cosx=eix+eix2.

Similarly, simplify the expression eixeix by the use of Euler’s formula as follows.

eixeix=(cosx+isinx)(cos(x)+isin(x))=cosx+isinxcosx+isinx[cos(x)=cosxandsin(x)=sinx]=isinx+isinx=2isinx

Thus, sinx=eixeix2i.

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