Download 9-5 Complex Numbers and De Moivre`s Theorem

9-5 Complex Numbers and De Moivre's Theorem
Graph each number in the complex plane and find its absolute value.
7. z = –3 – 7i
SOLUTION: For z = –3 – 7i, (a, b) = (–3, –7). Graph the point (–3, –7) in the complex plane.
Use the absolute value of a complex number formula.
Express each complex number in polar form.
11. –2 + i
SOLUTION: –2 + i
Find the modulus r and argument
.
The polar form of −2 + i is
(cos 2.68 + i sin 2.68).
13. 2 – 2i
SOLUTION: 2 – 2i
Find the modulus r and argument
.
The polar form of 2 – 2i is
15. –2 + 4i
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SOLUTION: –2 + 4i
.
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9-5 The
Complex
Numbers
polar form
of 2 – 2i is and De Moivre's Theorem
.
15. –2 + 4i
SOLUTION: –2 + 4i
Find the modulus r and argument
.
The polar form of −2 + 4i is 2
(cos 2.03 + i sin 2.03).
17. 3 + 3i
SOLUTION: 3 + 3i
Find the modulus r and argument
.
The polar form of 3+ 3i is
.
Graph each complex number on a polar grid. Then express it in rectangular form.
19. 2(cos 3 + i sin 3)
SOLUTION: The value of r is 2, and the value of
than
but less than π.
is 3. Plot the polar coordinates (2, 3). Notice that 3 radians is slightly greater To express the number in rectangular form, evaluate the trigonometric values and simplify.
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The rectangular form of 2(cos 3 + i sin 3) is −1.98 + 0.28i.
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9-5 The
Complex
Numbers
polar form
of 3+ 3i is and De Moivre's. Theorem
Graph each complex number on a polar grid. Then express it in rectangular form.
19. 2(cos 3 + i sin 3)
SOLUTION: The value of r is 2, and the value of
than
but less than π.
is 3. Plot the polar coordinates (2, 3). Notice that 3 radians is slightly greater To express the number in rectangular form, evaluate the trigonometric values and simplify.
The rectangular form of 2(cos 3 + i sin 3) is −1.98 + 0.28i.
21. SOLUTION: The value of r is 3, and the value of
is . Plot the polar coordinates
.
To express the number in rectangular form, evaluate the trigonometric values and simplify.
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The rectangular form of
is
i.
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To express the number in rectangular form, evaluate the trigonometric values and simplify.
9-5 The
Complex
Numbers
and3 +De
i sinMoivre's
rectangular
form of 2(cos
3) is −1.98 +Theorem
0.28i.
21. SOLUTION: The value of r is 3, and the value of
is . Plot the polar coordinates
.
To express the number in rectangular form, evaluate the trigonometric values and simplify.
The rectangular form of
is
i.
23. SOLUTION: The value of r is 2, and the value of
is . Plot the polar coordinates
.
To express the number in rectangular form, evaluate the trigonometric values and simplify.
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Page 4
i.
9-5 The
Complex
Numbers
rectangular
form of and De Moivre's
is Theorem
23. SOLUTION: The value of r is 2, and the value of
is . Plot the polar coordinates
.
To express the number in rectangular form, evaluate the trigonometric values and simplify.
is−1 −
The rectangular form of
25. i.
(cos 360º + i sin 360º)
SOLUTION: The value of r is
, and the value of
is 360°. Plot the polar coordinates To express the number in rectangular form, evaluate the trigonometric values and simplify.
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The rectangular form of
(cos 360º + i sin 360º) is
.
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rectangularNumbers
form of
i.
is−1 −Theorem
9-5 The
Complex
and De Moivre's
25. (cos 360º + i sin 360º)
SOLUTION: The value of r is
, and the value of
is 360°. Plot the polar coordinates To express the number in rectangular form, evaluate the trigonometric values and simplify.
The rectangular form of
(cos 360º + i sin 360º) is
.
Find each product or quotient and express it in rectangular form.
27. 5(cos 135° + i sin 135°) ⋅ 2 (cos 45° + i sin 45°)
SOLUTION: Use the Product Formula to find the product in polar form.
Now find the rectangular form of the product.
The polar form is 10(cos 180° + i sin 180°). The rectangular form is −10.
29. 2(cos 90º + i sin 90º) ⋅ 2(cos 270º + i sin 270º)
SOLUTION: Use the Product Formula to find the product in polar form.
NowManual
find the
rectangular
form
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by Cognero
of the product.
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Now find the rectangular form of the product.
9-5 The
Complex
Numbers
and
Moivre's
Theorem
polar form
is 10(cos 180°
+ iDe
sin 180°).
The rectangular
form is −10.
29. 2(cos 90º + i sin 90º) ⋅ 2(cos 270º + i sin 270º)
SOLUTION: Use the Product Formula to find the product in polar form.
Now find the rectangular form of the product.
The polar form is
. The rectangular form is 4.
31. SOLUTION: Use the Quotient Formula to find the quotient in polar form.
Now find the rectangular form.
The polar form of the quotient is
. The rectangular form of the quotient is
.
33. SOLUTION: Use the Quotient Formula to find the quotient in polar form.
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Now find the rectangular form.
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9-5 The
Complex
Numbers
andis De Moivre's Theorem
polar form
of the quotient
. The rectangular form of the quotient is
.
33. SOLUTION: Use the Quotient Formula to find the quotient in polar form.
Now find the rectangular form.
The polar form of the quotient is
. The rectangular form of the quotient is
.
35. SOLUTION: Use the Quotient Formula to find the quotient in polar form.
Now find the rectangular form of the product.
The polar form of the quotient is
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. The rectangular form of the quotient is
.
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