Mathematical Formulas for Industrial and Mechanical Engineering

Since

π

4

1

π

6

5

10π

24

5

5π

12

and

z

1

z

2

5

ffiffiffi

2

p

ðcosðπ=4Þ1 i sinðπ=4ÞÞ

2ðcosðπ=6Þ1 i sinðπ=6ÞÞ

5

ffiffiffi

2

p

2

cos

π

12

1 i sin

π

12



We can use z

2

5 z  z 5 r  rðcosðθ 1 θÞ1 i sinðθ 1 θÞÞ5 r

2

ðcos 2 θ 1 i sin 2θÞ

2.15 DeMoivre’s Theorem

z

n

5 r

n

ðcos n θ 1 i sin nθÞ

where n is a positive integer.

Let r 5 1 to getðcos θ1i sin θÞ

n

5 cos nθ 1 i sin nθ.

Example Compute ð11iÞ

6

ð11iÞ

6

5

ffiffiffi

2

p

cos

π

4

1i sin

π

4



6

5

ffiffiffi

2

p

6

cos 6 

π

4

1 i sin 6 

π

4

0

@

1

A

5 8 cos

3π

2

1 i sin

3π

2

0

@

1

A

528i

2.16 Euler’s Formula

e

ix

5 cos x 1 i sin x:

30 Mathematical Formulas for Industrial and Mechanical Engineering