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6.5 Trigonometric Form of a Complex number
De Moivre’s theorem
Graph the Complex number 2+3i
On a Complex Plane
real
imaginary
Absolute Value of the Complex number 4+3i
On a Complex Plane
real
imaginary
53434 22
22
i
babia
Trigonometric Form of a Complex Number
Given: a + bi
r is called the Modulus
θ is the Argument
r
sincos irz
22 bar
abtan
sin,cos rir
Writing a Complex number in Trigonometric form
Point: sin,cos rir
iz 54
Writing a Complex number in Trigonometric form
Point: sin,cos rir
415454
54
22
ir
iz
Writing a Complex number in Trigonometric form
Point: sin,cos rir
7.1283.511803.51
4
5tan
4
5tan
415454
54
1
22
so
ir
iz
3.51
7.128
Writing a Complex number in Trigonometric form
Point: sin,cos rir
7.1283.511803.51
4
5tan
4
5tan
415454
54
1
22
so
ir
iz
7.12841
7.1287.12841
7.128417.12841
cisz
iSinCosz
SinCosz
Product and Quotient of two Complex Numbers
02
2121
2
1
2
1
21212121
2222
1111
z
iSinCosr
r
z
z
iSinCosrrzz
iSinCosrz
iSinCosrz
Find the product
4545cos300
451520
2815
1720
21
2
1
iSin
ciszz
cisz
cisz
Find the product
2150215022
22
300
4545cos300
451520
2815
1720
21
2
1
ii
iSin
ciszz
cisz
cisz
Find the quotient
ii
iSin
ciszz
cisz
cisz
24.5748.294)1908.0(9816.0300
)11()11cos(300
)2817(1520
2815
1720
21
2
1
Power of Complex Numbers
De Moivre’s Theorem
niSinnCosrz
iSinCosrz
iSinCosrz
nn
2222
Power of Complex Numbers
Solve
135216
1354
2
cisz
cisz
Power of Complex Numbers
Solve
iiz
iSinCosz
cisz
cisz
161016
27027016
135216
1354
2
2
2
N th Roots of a Complex Number
Where k = 0, 1, 2, …., n - 1
nk
iSinnk
Cosrn 22
Find the Cube roots of 1000
K = 0, 1, 2..(3-1) done
355
322
322
10
355312
3)1(2
10
10302
302
10
32
32
10003
iiSinCos
iiSinCos
iSinCos
kiSin
kCos
Homework
Page 457 – 459
# 1, 9, 17, 25, 33,
41, 49, 57, 65,
77, 85, 93, 101,
109
Homework
Page 457 – 459 # 5, 13, 21, 29,
37, 45, 53, 61, 73, 81, 89, 97, 105, 113
“I got the Power!!!”