Download Section 5

Find rectangular coordinates for the point
with the given polar coordinates.
A.
B.
C.
D. (0, 3)
Find rectangular coordinates for the point
with the given polar coordinates.
A.
B.
C.
D. (0, 3)
You performed operations with complex numbers
written in rectangular form. (Lesson 0-6)
• Convert complex numbers from rectangular to polar
form and vice versa.
• Find products, quotients, powers, and roots of
complex numbers in polar form.
• complex plane
• trigonometric form
• real axis
• modulus
• imaginary axis
• argument
• Argand plane
• pth roots of unity
• absolute value of a
complex number
• polar form
Graphs and Absolute Values of Complex
Numbers
A. Graph z = 2 + 3i in the complex plane and find
its absolute value.
(a, b) = (2, 3)
Graphs and Absolute Values of Complex
Numbers
Absolute value formula
a = 2 and b = 3
Simplify.
The absolute value of 2 + 3i is
Answer:
Graphs and Absolute Values of Complex
Numbers
Absolute value formula
a = 2 and b = 3
Simplify.
The absolute value of 2 + 3i is
Answer:
Graphs and Absolute Values of Complex
Numbers
B. Graph z = –3 + i in the complex plane and find
its absolute value.
(a, b) = (–3, 1)
Graphs and Absolute Values of Complex
Numbers
Absolute value formula
a = –3 and b = 1
Simplify.
The absolute value of –3 + i is
Answer:
Graphs and Absolute Values of Complex
Numbers
Absolute value formula
a = –3 and b = 1
Simplify.
The absolute value of –3 + i is
Answer:
Graph 3 – 4i in the complex plane and find its
absolute value.
A. 5;
C. 1;
B. 5;
D. 7;
Graph 3 – 4i in the complex plane and find its
absolute value.
A. 5;
C. 1;
B. 5;
D. 7;
Complex Numbers in Polar Form
A. Express the complex number –2 + 5i in polar
form.
Find the modulus r and argument .
Conversion
formula
a = –2 and b = 5
Simplify.
The polar form of –2 + 5i is about
5.39(cos 1.95 + i sin 1.95).
Answer:
Complex Numbers in Polar Form
A. Express the complex number –2 + 5i in polar
form.
Find the modulus r and argument .
Conversion
formula
a = –2 and b = 5
Simplify.
The polar form of –2 + 5i is about
5.39(cos 1.95 + i sin 1.95).
Answer: 5.39(cos 1.95 + i sin 1.95)
Complex Numbers in Polar Form
B. Express the complex number 6 + 2i in polar form.
Find the modulus r and argument .
Conversion
formula
a = 6 and b = 2
Simplify.
The polar form of 6 + 2i is about
6.32(cos 0.32 +i sin 0.32).
Answer:
Complex Numbers in Polar Form
B. Express the complex number 6 + 2i in polar form.
Find the modulus r and argument .
Conversion
formula
a = 6 and b = 2
Simplify.
The polar form of 6 + 2i is about
6.32(cos 0.32 +i sin 0.32).
Answer: 6.32(cos 0.32 + i sin 0.32)
Express the complex number 4 – 5i in polar form.
A. 20(cos 5.61 + i sin 5.61)
B. 20(cos 0.90 + i sin 0.90)
C. 6.40(cos 4.04 + i sin 4.04)
D. 6.40(cos 5.39 + i sin 5.39)
Express the complex number 4 – 5i in polar form.
A. 20(cos 5.61 + i sin 5.61)
B. 20(cos 0.90 + i sin 0.90)
C. 6.40(cos 4.04 + i sin 4.04)
D. 6.40(cos 5.39 + i sin 5.39)
Graph and Convert the Polar Form of a
Complex Number
Graph
on a polar grid. Then
express it in rectangular form.
The value of r is 4, and the value of  is
Plot the polar coordinates
Graph and Convert the Polar Form of a
Complex Number
To express the number in rectangular form, evaluate
the trigonometric values and simplify.
Polar form
Evaluate for cosine and sine.
Distributive Property
The rectangular form of
Graph and Convert the Polar Form of a
Complex Number
Answer:
Graph and Convert the Polar Form of a
Complex Number
Answer:
Express
A. –6 – 6i
B.
C.
D.
in rectangular form.
Express
A. –6 – 6i
B.
C.
D.
in rectangular form.
Product of Complex Numbers in Polar Form
Find
in polar
form. Then express the product in rectangular
form.
Original
expression
Product Formula
Simplify.
Product of Complex Numbers in Polar Form
Now find the rectangular form of the product.
10(cos π + i sin π)
Polar form
= 10(–1 + 0i)
Evaluate.
= –10 + 0i
Distributive Property
The polar form of the product is 10(cos π + i sin π).
The rectangular form of the product is –10 + 0i or –10.
Answer:
Product of Complex Numbers in Polar Form
Now find the rectangular form of the product.
10(cos π + i sin π)
Polar form
= 10(–1 + 0i)
Evaluate.
= –10 + 0i
Distributive Property
The polar form of the product is 10(cos π + i sin π).
The rectangular form of the product is –10 + 0i or –10.
Answer: 10(cos π + i sin π); –10
Find
your answer in rectangular form.
A. –7.25 + 27.05i
B. –19.80 – 19.80i
C. –27.05 + 7.25i
D. –10.63 + 2.85i
Express
Find
your answer in rectangular form.
A. –7.25 + 27.05i
B. –19.80 – 19.80i
C. –27.05 + 7.25i
D. –10.63 + 2.85i
Express
Quotient of Complex Numbers
in Polar Form
ELECTRICITY If a circuit has a voltage E of 100
volts and an impedance Z of 4 – 3j ohms, find the
current I in the circuit in rectangular form.
Use E = I • Z.
Express each number in polar form.
100 = 100(cos 0 +j sin 0)
4 – 3j = 5[cos (–0.64) + jsin (–0.64)]
Quotient of Complex Numbers
in Polar Form
Solve for the current I in E = I • Z.
I•Z=E
Original equation
Divide each side by Z.
E = 100(cos 0 + j sin 0)
Z = 5[cos (–0.64) + j sin (–0.64)]
Quotient of Complex Numbers
in Polar Form
Quotient Formula
Simplify.
Now, convert the current to rectangular form.
I = 20(cos 0.64 + j sin 0.64)
Original equation
= 20(0.80 + 0.60j)
Evaluate.
= 16.04 + 11.94j
Distributive Property
The current is about 16.04 + 11.94j amps.
Answer:
Quotient of Complex Numbers
in Polar Form
Quotient Formula
Simplify.
Now, convert the current to rectangular form.
I = 20(cos 0.64 + j sin 0.64)
Original equation
= 20(0.80 + 0.60j)
Evaluate.
= 16.04 + 11.94j
Distributive Property
The current is about 16.04 + 11.94j amps.
Answer: 16.04 + 11.94j amps
ELECTRICITY If a circuit has a voltage of 140 volts
and a current of 4 + 3j amps, find the impedance of
the circuit in rectangular form.
A. 0.03 + 0.02j
B. 22.4 – 16.8j
C. 560 + 420j
D. 23.4 + 16.87j
ELECTRICITY If a circuit has a voltage of 140 volts
and a current of 4 + 3j amps, find the impedance of
the circuit in rectangular form.
A. 0.03 + 0.02j
B. 22.4 – 16.8j
C. 560 + 420j
D. 23.4 + 16.87j
De Moivre’s Theorem
Find
First, write
and express in rectangular form.
in polar form.
Conversion formula
a = 3 and b =
Simplify.
Simplify.
De Moivre’s Theorem
The polar form of
is
Now use De Moivre's Theorem to find the fourth
power.
Original
equation
De Moivre's
Theorem
Simplify.
De Moivre’s Theorem
Evaluate.
Simplify.
Therefore,
Answer:
De Moivre’s Theorem
Evaluate.
Simplify.
Therefore,
Answer:
Find
A. 1728i
B. 1728
C.
D.
and express in rectangular form.
Find
A. 1728i
B. 1728
C.
D.
and express in rectangular form.
• complex plane
• trigonometric form
• real axis
• modulus
• imaginary axis
• argument
• Argand plane
• pth roots of unity
• absolute value of a
complex number
• polar form