LESSON 9-5: COMPLEX NUMBERS AND DE MOIVRE’S THEOREM  Conver complex numbers from rectangular to polar form and vice versa.  Find products, quotients, powers, and roots of complex numbers in polar form.

        

complex plane real axis imaginary axis Argand plane absolute value of a complex number polar form/ trigonometric form modulus argument pth roots of unity

Graph each number in the complex plane, and find its absolute value.

A. 5  2i

B. 3  2i

C.  3  4i

D.  5  3i

 THE ANGLE θ CAN ALWAYS BE REWRITTEN SO THAT 0 ≤ θ ≤ 2π.  Shorthand for polar/trigonometric form is z = 𝒓𝒄𝒊𝒔𝜽.

Express each complex number in polar form.

A. 9  7i

B.  2  5i

C.  6  6i

D. 2  8i

Graph each complex number on a polar grid. Then express it in rectangular form.

2 2  A. 4  cos  i sin 3 3 

2   4 cis  3 

11 11  B. 5  cos  i sin 6 6 

11   5 cis  6 

THE ANGLE θ CAN ALWAYS BE REWRITTEN SO THAT 0 ≤ θ ≤ 2π.  Shorthand forms are 𝒛𝟏 𝒛𝟐 = 𝒓𝟏 𝒓𝟐 𝒄𝒊𝒔(𝜽𝟏 + 𝜽𝟐 ) and

z1 r1  z2 r2

𝒄𝒊𝒔(𝜽𝟏 − 𝜽𝟐 ).

Find each product in polar form. Then express the product in rectangular form.

4 4   A. 3  cos  i sin 3 3  

3 3   B.  6  cos  i sin 4 4  

7 7   5  cos  i sin 4 4  

2 2   2  cos  i sin 3 3  

Express answers for each of the following in polar and rectangular forms. 2 2  2 3 3  3   A. Find z1  z2 if z1  2  cos  i sin  2cis and z2  6  cos  i sin  6cis   3 3  3 4 4  4  

1 3 3  B. 3  cos   i sin     cos  i sin 2 4 4 

C. ELECTRICITY If a circuit has a voltage E of 170 volts and an impedance Z of 5 – 2j ohms, find the current I amps in the circuit. Use formula E = I·Z.

Shorthand form is z 𝒏 = 𝒓𝒏 𝒄𝒊𝒔(𝒏𝜽).

Find each power, and express it in rectangular form.

A.

 6  6 3i 

B.

2

3  2i



8

3

C.

 3  5i 

5

 Shorthand form is

   2n  r p cis  p   1

where 𝒏 = 𝟎, 𝟏, 𝟐, … , 𝒑 − 𝟏.

A. Find and graph the fifth roots of  2  2i .

B. Find and graph the cube roots of 3  3 3i .

The pth Roots of a Complex Number Each root will have modulus of argument of the first root is adding

2 p

 , p

1 p

r .

The

and each successive root is found by repeatedly

to the argument.

Find and graph the fourth roots of unity.