DeMoivre's Theorem, Trigonometry

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Post by varsha sharma on June 9, 2011

I want to master your notes and use them in my classroom and i want to work as an instructor in school of math and science.
Thanks,
Varsha

DeMoivre's Theorem

Main formulas:

If the complex number z is written in polar form z = re^iθ, then we can find n-th powers as follows:

zⁿ = (re^iθ)ⁿ = [r(cosθ + i sinθ )]ⁿ = rⁿ(cosn θ + i sinn θ ) = rⁿ e^{i nθ}
Every nonzero complex number has exactly n n-th roots.
We can find n-th roots as follows:

n

√

z

=

z^[1/(n)] = (re^iθ)^[1/(n)]

=

[r(cosθ + i sinθ )]^[1/(n)]

=

r^[1/(n)] ( cos θ + 2kπ
n
+ i sin θ + 2kπ
n
)

=

r^[1/(n)] e^{i [(θ
+ 2kπ)/n]}

where k = 0,1,2,..., n− 1.

Example 1:

Convert the complex number z = − √ 3 + i into polar form and then use DeMoivre's Theorem to calculate z⁷.

Example 2:

Find all complex eighth roots of 16.

Example 3:

Find all complex cube roots of − 1.

Example 4:

Convert the complex number z = 2√ 2 − 2√ 2i into polar form and then use DeMoivre's Theorem to calculate z⁵.

Example 5:

Find all complex fourth roots of z = − 2 − 2√ 3i.

Give me another blank page here.

Convert the complex number w = − 1 + √3 i into polar form and then use DeMoivre's Theorem to calculate w⁵

DeMoivre's Theorem: zⁿ = (re^iθ)ⁿ = [ r(cosθ+ isinθ) ]ⁿ = rⁿ(cosnθ+ isinnθ) = rⁿe^inθ
r = √{x² + y²}
r = √{( − 1)² + (√3 )²} = 2
θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
θ = arctan([(√3 )/( − 1)]) + π = − [(π)/3] + π = [(2π)/3]
w⁵ = (2e^{^[(2π)/3]i})⁵ = 2⁵e^{^[(10π)/3]i} = 32e^{^[(4π)/3]i}
32(cos[(4π)/3] + isin[(4π)/3]) = 32[ ( − [1/2]) + i([( − √3 )/2]) ]

− 16 − 16√{3 i}

Convert the complex number w = − 4√3 + 4i into polar form and then use DeMoivre's Theorem to calculate w⁶

DeMoivre's Theorem: zⁿ = (re^iθ)ⁿ = [ r(cosθ+ isinθ) ]ⁿ = rⁿ(cosnθ+ isinnθ) = rⁿe^inθ
r = √{x² + y²}
r = √{( − 4√3 )² + (4)²} = 8
θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
θ = arctan([4/( − 4√3 )]) + π = − [(π)/6] + π = [(5π)/6]
w⁶ = (8e^{^[(5π)/6]i})⁶ = 8⁶e^{^[(30π)/6]i} = 262144e^5πi = 262144e^πi
262144(cosπ+ isinπ) = 262144[ ( − 1) + i(0) ]

− 262144

Convert the complex number w = − 2 + 2√{3 i} into polar form and then use DeMoivre's Theorem to calculate w⁷

DeMoivre's Theorem: zⁿ = (re^iθ)ⁿ = [ r(cosθ+ isinθ) ]ⁿ = rⁿ(cosnθ+ isinnθ) = rⁿe^inθ
r = √{x² + y²}
r = √{( − 2)² + (2√3 )²} = 4
θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
θ = arctan([(2√3 )/( − 2)]) + π = − [(π)/3] + π = [(2π)/3]
w⁷ = (4e^{^[(2π)/3]i})⁷ = 4⁷e^{^[(14π)/3]i} = 16384e^{^[(2π)/3]i}
16384(cos[(2π)/3] + isin[(2π)/3]) = 16384[ ( − [1/2]) + i([(√3 )/2]) ]

− 8192 + 8192√{3i}

Convert the complex number w = − 3√2 − 3√{2i} into polar form and then use DeMoivre's Theorem to calculate w⁴

DeMoivre's Theorem: zⁿ = (re^iθ)ⁿ = [ r(cosθ+ isinθ) ]ⁿ = rⁿ(cosnθ+ isinnθ) = rⁿe^inθ
r = √{x² + y²}
r = √{( − 3√2 )² + ( − 3√2 )²} = 6
θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
θ = arctan([( − 3√2 )/( − 3√2 )]) + π = [(π)/4] + π = [(5π)/4]
w⁴ = (6e^{^[(5π)/4]i})⁴ = 6⁴e^5πi = 1296e^5πi = 1296e^πi
1296(cosπ+ isinπ) = 1296[ ( − 1) + i(0) ]

− 1296

Convert the complex number w = 4 + 4i into polar form and then use DeMoivre's Theorem to calculate w³

DeMoivre's Theorem: zⁿ = (re^iθ)ⁿ = [ r(cosθ+ isinθ) ]ⁿ = rⁿ(cosnθ+ isinnθ) = rⁿe^inθ
r = √{x² + y²}
r = √{(4)² + (4)²} = 4√2
θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
θ = arctan([4/4]) = [(π)/4]
w³ = (4√2 e^{^[(π)/4]i})⁴ = (4√2 )³e^{^[(3π)/4]i} = 128√2 e^{^[(3π)/4]i}
128√2 (cos[3p/4] + isin[3p/4]) = 16384[ ( − [(√2 )/2]) + i([(√2 )/2]) ]

− 128 + 128i

Convert the complex number w = 2 − 2√{3 i} into polar form and then use DeMoivre's Theorem to calculate w⁴

DeMoivre's Theorem: zⁿ = (re^iθ)ⁿ = [ r(cosθ+ isinθ) ]ⁿ = rⁿ(cosnθ+ isinnθ) = rⁿe^inθ
r = √{x² + y²}
r = √{(2)² + ( − 2√3 )²} = 4
θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
θ = arctan([( − 2√3 )/2]) = − [(π)/3] + 2π = [(5π)/3]
w⁴ = (4e^{^[(5π)/3]i})⁴ = 4⁴e^{^[(20π)/3]i} = 256e^{^[(2π)/3]i}
256(cos[(2π)/3] + isin[(2π)/3]) = 256[ ( − [1/2]) + i([(√3 )/2]) ]

− 128 + 128√{3i}

Find all sixth roots of 1

There should be six different answers
DeMoivre's Theorem: z^{^[1/n]} = (re^iθ)^{^[1/n]} = [ r(cosθ+ isinθ) ]^{^[1/n]} = r^{^[1/n]}(cos[(θ+ 2kπ)/n] + isin[(θ+ 2kπ)/n]) = r^{^[1/n]}e^{^{i[(θ+ 2kπ)/n]}}
r = √{x² + y²}
θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
1 + 0i
r = √{(1)² + (0)²} = 1
θ = arctan([0/1]) = 0
z = 1eⁱ⁰ ⇒ z^[1/6] = 1^[1/6](cos[(0 + 2kπ)/6] + isin[(0 + 2kπ)/6])

k [(θ+ 2kπ)/n] = [(0 + 2kπ)/6] = [(kπ)/3] cosα+i sinα Six Answers

0 0 1+0i 1

1 [(π)/3] [1/2] + [(√3 )/2]i [1/2] + [(√3 )/2]i

2 [(2π)/3] − [1/2] + [(√3 )/2]i − [1/2] + [(√3 )/2]i

3 π − 1 + 0i −1

4 [(4π)/3] − [1/2] − [(√3 )/2]i − [1/2] − [(√3 )/2]i

Find all fourth roots of - 4

There should be four different answers
DeMoivre's Theorem: z^{^[1/n]} = (re^iθ)^{^[1/n]} = [ r(cosθ+ isinθ) ]^{^[1/n]} = r^{^[1/n]}(cos[(θ+ 2kπ)/n] + isin[(θ+ 2kπ)/n]) = r^{^[1/n]}e^{^{i[(θ+ 2kπ)/n]}}
r = √{x² + y²}
θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
− 4 + 0i
r = √{( − 4)² + (0)²} = 4
θ = arctan([0/( − 4)]) + π = π
z = 4e^iπ ⇒ z^{^[1/4]} = 1^{^[1/4]}(cos[(π+ 2kπ)/4] + isin[(π+ 2kπ)/4])

k [(θ+ 2kπ)/n] = [(π+ 2kπ)/4] r^{^[1/n]}(cosα+ isinα) Four Answers

0 [(π)/4] √2 ([(√2 )/2] + i[(√2 )/2]) 1+i

1 [(3π)/4] √2 ( − [(√2 )/2] + i[(√2 )/2]) −1+i

2 [(5π)/4] √2 ( − [(√2 )/2] + i( − [(√2 )/2]) ) −1−i

3 [(7π)/4] √2 ( [(√2 )/2] + i( − [(√2 )/2]) ) 1−i

Find all cube roots of 8

There should be three different answers
DeMoivre's Theorem: z^{^[1/n]} = (re^iθ)^{^[1/n]} = [ r(cosθ+ isinθ) ]^{^[1/n]} = r^{^[1/n]}(cos[(θ+ 2kπ)/n] + isin[(θ+ 2kπ)/n]) = r^{^[1/n]}e^{^{i[(θ+ 2kπ)/n]}}
r = √{x² + y²}
θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
8 + 0i
r = √{(8)² + (0)²} = 1
θ = arctan([0/8]) = 0
z = 8eⁱ⁰ ⇒ z^[1/3] = 8^[1/3](cos[(0 + 2kπ)/3] + isin[(0 + 2kπ)/3])

k [(θ+ 2kπ)/n] = [(0 + 2kπ)/3] 2(cosα+ isinα) Three Answers

0 0 2(1 + 0i) 2

1 [(2π)/3] 2( − [1/2] + i[(√3 )/2] ) − 1 + √3 i

2 [(4π)/3] 2( − [1/2] − i[(√3 )/2] ) − 1 − √3 i

Find all cube roots of -125

There should be three different answers
DeMoivre's Theorem: z^{^[1/n]} = (re^iθ)^{^[1/n]} = [ r(cosθ+ isinθ) ]^{^[1/n]} = r^{^[1/n]}(cos[(θ+ 2kπ)/n] + isin[(θ+ 2kπ)/n]) = r^{^[1/n]}e^{^{i[(θ+ 2kπ)/n]}}
r = √{x² + y²}
θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
− 125 + 0i
r = √{( − 125)² + (0)²} = 125
θ = arctan([0/( − 125)]) + π = π
z = 125e^iπ ⇒ z^[1/3] = 125^[1/3](cos[(π+ 2kπ)/3] + isin[(π+ 2kπ)/3])

k [(θ+ 2kπ)/n] = [(π+ 2kπ)/3] 5(cosα+ isinα) Three Answers

0 [(π)/3] 5( [1/2] + i[(√3 )/2] ) [5/2] + [(5√3 )/2]i

1 π 5( − 1 + i(0) ) −5

2 [(5π)/3] 5( [1/2] + i( − [(√3 )/2]) ) [5/2] − [(5√3 )/2]i

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*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

DeMoivre's Theorem

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Intro

Introduction to DeMoivre's Theorem

n nth Roots

DeMoivre's Theorem: Finding nth Roots

Example 1: Convert to Polar Form and Use DeMoivre's Theorem

Example 2: Find Complex Eighth Roots

Example 3: Find Complex Roots

Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem

Extra Example 2: Find Complex Fourth Roots

I. Trigonometric Functions
	Angles	39:05
	Sine and Cosine Functions	43:16
	Sine and Cosine Values of Special Angles	33:05
	Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D	52:03
	Tangent and Cotangent Functions	36:04
	Secant and Cosecant Functions	27:18
	Inverse Trigonometric Functions	32:58
	Computations of Inverse Trigonometric Functions	31:08
II. Trigonometric Identities
	Pythagorean Identity	19:11
	Identity Tan(squared)x+1=Sec(squared)x	23:16
	Addition and Subtraction Formulas	52:52
	Double Angle Formulas	29:05
	Half-Angle Formulas	43:55
III. Applications of Trigonometry
	Trigonometry in Right Angles	25:43
	Law of Sines	56:40
	Law of Cosines	49:05
	Finding the Area of a Triangle	27:37
	Word Problems and Applications of Trigonometry	34:25
	Vectors	46:42
IV. Complex Numbers and Polar Coordinates
	Polar Coordinates	67:35
	Complex Numbers	35:59
	Polar Form of Complex Numbers	40:43
	DeMoivre's Theorem	57:37

Mathematics: Trigonometry

I. Trigonometric Functions
	Angles			39:05
		Intro		0:00
		Degrees		0:22
			Circle is 360 Degrees	0:48
			Splitting a Circle	1:13
		Radians		2:08
			Circle is 2 Pi Radians	2:31
			One Radian	2:52
			Half-Circle and Right Angle	4:00
		Converting Between Degrees and Radians		6:24
			Formulas for Degrees and Radians	6:52
		Coterminal, Complementary, Supplementary Angles		7:23
			Coterminal Angles	7:30
			Complementary Angles	9:40
			Supplementary Angles	10:08
		Example 1: Dividing a Circle		10:38
		Example 2: Converting Between Degrees and Radians		11:56
		Example 3: Quadrants and Coterminal Angles		14:18
		Extra Example 1: Common Angle Conversions		8:02
		Extra Example 2: Quadrants and Coterminal Angles		7:14
	Sine and Cosine Functions			43:16
		Intro		0:00
		Sine and Cosine		0:15
			Unit Circle	0:22
			Coordinates on Unit Circle	1:03
			Right Triangles	1:52
			Adjacent, Opposite, Hypotenuse	2:25
			Master Right Triangle Formula: SOHCAHTOA	2:48
		Odd Functions, Even Functions		4:40
			Example: Odd Function	4:56
			Example: Even Function	7:30
		Example 1: Sine and Cosine		10:27
		Example 2: Graphing Sine and Cosine Functions		14:39
		Example 3: Right Triangle		21:40
		Example 4: Odd, Even, or Neither		26:01
		Extra Example 1: Right Triangle		4:05
		Extra Example 2: Graphing Sine and Cosine Functions		5:23
	Sine and Cosine Values of Special Angles			33:05
		Intro		0:00
		45-45-90 Triangle and 30-60-90 Triangle		0:08
			45-45-90 Triangle	0:21
			30-60-90 Triangle	2:06
		Mnemonic: All Students Take Calculus (ASTC)		5:21
			Using the Unit Circle	5:59
			New Angles	6:21
			Other Quadrants	9:43
			Mnemonic: All Students Take Calculus	10:13
		Example 1: Convert, Quadrant, Sine/Cosine		13:11
		Example 2: Convert, Quadrant, Sine/Cosine		16:48
		Example 3: All Angles and Quadrants		20:21
		Extra Example 1: Convert, Quadrant, Sine/Cosine		4:15
		Extra Example 2: All Angles and Quadrants		4:03
	Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D			52:03
		Intro		0:00
		Amplitude and Period of a Sine Wave		0:38
			Sine Wave Graph	0:58
			Amplitude: Distance from Middle to Peak	1:18
			Peak: Distance from Peak to Peak	2:41
		Phase Shift and Vertical Shift		4:13
			Phase Shift: Distance Shifted Horizontally	4:16
			Vertical Shift: Distance Shifted Vertically	6:48
		Example 1: Amplitude/Period/Phase and Vertical Shift		8:04
		Example 2: Amplitude/Period/Phase and Vertical Shift		17:39
		Example 3: Find Sine Wave Given Attributes		25:23
		Extra Example 1: Amplitude/Period/Phase and Vertical Shift		7:27
		Extra Example 2: Find Cosine Wave Given Attributes		10:27
	Tangent and Cotangent Functions			36:04
		Intro		0:00
		Tangent and Cotangent Definitions		0:21
			Tangent Definition	0:25
			Cotangent Definition	0:47
		Master Formula: SOHCAHTOA		1:01
			Mnemonic	1:16
		Tangent and Cotangent Values		2:29
			Remember Common Values of Sine and Cosine	2:46
			90 Degrees Undefined	4:36
		Slope and Mnemonic: ASTC		5:47
			Uses of Tangent	5:54
			Example: Tangent of Angle is Slope	6:09
			Sign of Tangent in Quadrants	7:49
		Example 1: Graph Tangent and Cotangent Functions		10:42
		Example 2: Tangent and Cotangent of Angles		16:09
		Example 3: Odd, Even, or Neither		18:56
		Extra Example 1: Tangent and Cotangent of Angles		2:27
		Extra Example 2: Tangent and Cotangent of Angles		5:02
	Secant and Cosecant Functions			27:18
		Intro		0:00
		Secant and Cosecant Definitions		0:17
			Secant Definition	0:18
			Cosecant Definition	0:33
		Example 1: Graph Secant Function		0:48
		Example 2: Values of Secant and Cosecant		6:49
		Example 3: Odd, Even, or Neither		12:49
		Extra Example 1: Graph of Cosecant Function		4:58
		Extra Example 2: Values of Secant and Cosecant		5:19
	Inverse Trigonometric Functions			32:58
		Intro		0:00
		Arcsine Function		0:24
			Restrictions between -1 and 1	0:43
			Arcsine Notation	1:26
		Arccosine Function		3:07
			Restrictions between -1 and 1	3:36
			Cosine Notation	3:53
		Arctangent Function		4:30
			Between -Pi/2 and Pi/2	4:44
			Tangent Notation	5:02
		Example 1: Domain/Range/Graph of Arcsine		5:45
		Example 2: Arcsin/Arccos/Arctan Values		10:46
		Example 3: Domain/Range/Graph of Arctangent		17:14
		Extra Example 1: Domain/Range/Graph of Arccosine		4:30
		Extra Example 2: Arcsin/Arccos/Arctan Values		5:40
	Computations of Inverse Trigonometric Functions			31:08
		Intro		0:00
		Inverse Trigonometric Function Domains and Ranges		0:31
			Arcsine	0:41
			Arccosine	1:14
			Arctangent	1:41
		Example 1: Arcsines of Common Values		2:44
		Example 2: Odd, Even, or Neither		5:57
		Example 3: Arccosines of Common Values		12:24
		Extra Example 1: Arctangents of Common Values		5:50
		Extra Example 2: Arcsin/Arccos/Arctan Values		8:51
II. Trigonometric Identities
	Pythagorean Identity			19:11
		Intro		0:00
		Pythagorean Identity		0:17
			Pythagorean Triangle	0:27
			Pythagorean Identity	0:45
		Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity		1:14
		Example 2: Find Angle Given Cosine and Quadrant		4:18
		Example 3: Verify Trigonometric Identity		8:00
		Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem		3:32
		Extra Example 2: Find Angle Given Cosine and Quadrant		3:55
	Identity Tan(squared)x+1=Sec(squared)x			23:16
		Intro		0:00
		Main Formulas		0:19
			Companion to Pythagorean Identity	0:27
			For Cotangents and Cosecants	0:52
			How to Remember	0:58
		Example 1: Prove the Identity		1:40
		Example 2: Given Tan Find Sec		3:42
		Example 3: Prove the Identity		7:45
		Extra Example 1: Prove the Identity		2:22
		Extra Example 2: Given Sec Find Tan		4:34
	Addition and Subtraction Formulas			52:52
		Intro		0:00
		Addition and Subtraction Formulas		0:09
			How to Remember	0:48
		Cofunction Identities		1:31
			How to Remember Graphically	1:44
			Where to Use Cofunction Identities	2:52
		Example 1: Derive the Formula for cos(A-B)		3:08
		Example 2: Use Addition and Subtraction Formulas		16:03
		Example 3: Use Addition and Subtraction Formulas to Prove Identity		25:11
		Extra Example 1: Use cos(A-B) and Cofunction Identities		7:54
		Extra Example 2: Convert to Radians and use Formulas		11:32
	Double Angle Formulas			29:05
		Intro		0:00
		Main Formula		0:07
			How to Remember from Addition Formula	0:18
			Two Other Forms	1:35
		Example 1: Find Sine and Cosine of Angle using Double Angle		3:16
		Example 2: Prove Trigonometric Identity using Double Angle		9:37
		Example 3: Use Addition and Subtraction Formulas		12:38
		Extra Example 1: Find Sine and Cosine of Angle using Double Angle		6:10
		Extra Example 2: Prove Trigonometric Identity using Double Angle		3:18
	Half-Angle Formulas			43:55
		Intro		0:00
		Main Formulas		0:09
			Confusing Part	0:34
		Example 1: Find Sine and Cosine of Angle using Half-Angle		0:54
		Example 2: Prove Trigonometric Identity using Half-Angle		11:51
		Example 3: Prove the Half-Angle Formula for Tangents		18:39
		Extra Example 1: Find Sine and Cosine of Angle using Half-Angle		7:16
		Extra Example 2: Prove Trigonometric Identity using Half-Angle		3:34
III. Applications of Trigonometry
	Trigonometry in Right Angles			25:43
		Intro		0:00
		Master Formula for Right Angles		0:11
			SOHCAHTOA	0:15
			Only for Right Triangles	1:26
		Example 1: Find All Angles in a Triangle		2:19
		Example 2: Find Lengths of All Sides of Triangle		7:39
		Example 3: Find All Angles in a Triangle		11:00
		Extra Example 1: Find All Angles in a Triangle		5:10
		Extra Example 2: Find Lengths of All Sides of Triangle		4:18
	Law of Sines			56:40
		Intro		0:00
		Law of Sines Formula		0:18
			SOHCAHTOA	0:27
			Any Triangle	0:59
			Graphical Representation	1:25
			Solving Triangle Completely	2:37
		When to Use Law of Sines		2:55
			ASA, SAA, SSA, AAA	2:59
			SAS, SSS for Law of Cosines	7:11
		Example 1: How Many Triangles Satisfy Conditions, Solve Completely		8:44
		Example 2: How Many Triangles Satisfy Conditions, Solve Completely		15:30
		Example 3: How Many Triangles Satisfy Conditions, Solve Completely		28:32
		Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely		8:01
		Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely		15:11
	Law of Cosines			49:05
		Intro		0:00
		Law of Cosines Formula		0:23
			Graphical Representation	0:34
			Relates Sides to Angles	1:00
			Any Triangle	1:20
			Generalization of Pythagorean Theorem	1:32
		When to Use Law of Cosines		2:26
			SAS, SSS	2:30
		Heron's Formula		4:49
			Semiperimeter S	5:11
		Example 1: How Many Triangles Satisfy Conditions, Solve Completely		5:53
		Example 2: How Many Triangles Satisfy Conditions, Solve Completely		15:19
		Example 3: Find Area of a Triangle Given All Side Lengths		26:33
		Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely		11:05
		Extra Example 2: Length of Third Side and Area of Triangle		9:17
	Finding the Area of a Triangle			27:37
		Intro		0:00
		Master Right Triangle Formula and Law of Cosines		0:19
			SOHCAHTOA	0:27
			Law of Cosines	1:23
		Heron's Formula		2:22
			Semiperimeter S	2:37
		Example 1: Area of Triangle with Two Sides and One Angle		3:12
		Example 2: Area of Triangle with Three Sides		6:11
		Example 3: Area of Triangle with Three Sides, No Heron's Formula		8:50
		Extra Example 1: Area of Triangle with Two Sides and One Angle		2:54
		Extra Example 2: Area of Triangle with Two Sides and One Angle		6:48
	Word Problems and Applications of Trigonometry			34:25
		Intro		0:00
		Formulas to Remember		0:11
			SOHCAHTOA	0:15
			Law of Sines	0:55
			Law of Cosines	1:48
			Heron's Formula	2:46
		Example 1: Telephone Pole Height		4:01
		Example 2: Bridge Length		7:48
		Example 3: Area of Triangular Field		14:20
		Extra Example 1: Kite Height		4:36
		Extra Example 2: Roads to a Town		10:34
	Vectors			46:42
		Intro		0:00
		Vector Formulas and Concepts		0:12
			Vectors as Arrows	0:28
			Magnitude	0:38
			Direction	0:50
			Drawing Vectors	1:16
			Uses of Vectors: Velocity, Force	1:37
			Vector Magnitude Formula	3:15
			Vector Direction Formula	3:28
			Vector Components	6:27
		Example 1: Magnitude and Direction of Vector		8:00
		Example 2: Force to a Box on a Ramp		12:25
		Example 3: Plane with Wind		18:30
		Extra Example 1: Components of a Vector		2:54
		Extra Example 2: Ship with a Current		13:13
IV. Complex Numbers and Polar Coordinates
	Polar Coordinates			67:35
		Intro		0:00
		Polar Coordinates vs Rectangular/Cartesian Coordinates		0:12
			Rectangular Coordinates, Cartesian Coordinates	0:23
			Polar Coordinates	0:59
		Converting Between Polar and Rectangular Coordinates		2:06
			R	2:16
			Theta	2:48
		Example 1: Convert Rectangular to Polar Coordinates		6:53
		Example 2: Convert Polar to Rectangular Coordinates		17:28
		Example 3: Graph the Polar Equation		28:00
		Extra Example 1: Convert Polar to Rectangular Coordinates		10:01
		Extra Example 2: Graph the Polar Equation		10:53
	Complex Numbers			35:59
		Intro		0:00
		Main Definition		0:07
			Number i	0:23
			Complex Number Form	0:33
		Powers of Imaginary Number i		1:00
			Repeating Pattern	1:43
		Operations on Complex Numbers		3:30
			Adding and Subtracting Complex Numbers	3:39
			Multiplying Complex Numbers	4:39
			FOIL Method	5:06
			Conjugation	6:29
		Dividing Complex Numbers		7:34
			Conjugate of Denominator	7:45
		Example 1: Solve For Complex Number z		11:02
		Example 2: Expand and Simplify		15:34
		Example 3: Simplify the Powers of i		17:50
		Extra Example 1: Simplify		4:37
		Extra Example 2: All Complex Numbers Satisfying Equation		10:00
	Polar Form of Complex Numbers			40:43
		Intro		0:00
		Polar Coordinates		0:49
			Rectangular Form	0:52
			Polar Form	1:25
			R and Theta	1:51
		Polar Form Conversion		2:27
			R and Theta	2:35
			Optimal Values	4:05
			Euler's Formula	4:25
		Multiplying Two Complex Numbers in Polar Form		6:10
			Multiply r's Together and Add Exponents	6:32
		Example 1: Convert Rectangular to Polar Form		7:17
		Example 2: Convert Polar to Rectangular Form		13:49
		Example 3: Multiply Two Complex Numbers		17:28
		Extra Example 1: Convert Between Rectangular and Polar Forms		6:48
		Extra Example 2: Simplify Expression to Polar Form		7:48
	DeMoivre's Theorem			57:37
		Intro		0:00
		Introduction to DeMoivre's Theorem		0:10
			n nth Roots	3:06
		DeMoivre's Theorem: Finding nth Roots		3:52
			Relation to Unit Circle	6:29
			One nth Root for Each Value of k	7:11
		Example 1: Convert to Polar Form and Use DeMoivre's Theorem		8:24
		Example 2: Find Complex Eighth Roots		15:27
		Example 3: Find Complex Roots		27:49
		Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem		7:41
		Extra Example 2: Find Complex Fourth Roots		14:36

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DeMoivre's Theorem

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DeMoivre's Theorem

Convert the complex number w = − 1 + √3 i into polar form and then use DeMoivre's Theorem to calculate w⁵

− 16 − 16√{3 i}

Convert the complex number w = − 4√3 + 4i into polar form and then use DeMoivre's Theorem to calculate w⁶

− 262144

Convert the complex number w = − 2 + 2√{3 i} into polar form and then use DeMoivre's Theorem to calculate w⁷

− 8192 + 8192√{3i}

Convert the complex number w = − 3√2 − 3√{2i} into polar form and then use DeMoivre's Theorem to calculate w⁴

− 1296

Convert the complex number w = 4 + 4i into polar form and then use DeMoivre's Theorem to calculate w³

− 128 + 128i

Convert the complex number w = 2 − 2√{3 i} into polar form and then use DeMoivre's Theorem to calculate w⁴

− 128 + 128√{3i}

Find all sixth roots of 1

k [(θ+ 2kπ)/n] = [(0 + 2kπ)/6] = [(kπ)/3] cosα+i sinα Six Answers

0 0 1+0i 1

1 [(π)/3] [1/2] + [(√3 )/2]i [1/2] + [(√3 )/2]i

2 [(2π)/3] − [1/2] + [(√3 )/2]i − [1/2] + [(√3 )/2]i

3 π − 1 + 0i −1

4 [(4π)/3] − [1/2] − [(√3 )/2]i − [1/2] − [(√3 )/2]i

Find all fourth roots of - 4

Find all cube roots of 8

k [(θ+ 2kπ)/n] = [(0 + 2kπ)/3] 2(cosα+ isinα) Three Answers

0 0 2(1 + 0i) 2

1 [(2π)/3] 2( − [1/2] + i[(√3 )/2] ) − 1 + √3 i

2 [(4π)/3] 2( − [1/2] − i[(√3 )/2] ) − 1 − √3 i

Find all cube roots of -125

k [(θ+ 2kπ)/n] = [(π+ 2kπ)/3] 5(cosα+ isinα) Three Answers

0 [(π)/3] 5( [1/2] + i[(√3 )/2] ) [5/2] + [(5√3 )/2]i

1 π 5( − 1 + i(0) ) −5

2 [(5π)/3] 5( [1/2] + i( − [(√3 )/2]) ) [5/2] − [(5√3 )/2]i

Answer

DeMoivre's Theorem

Mathematics: Trigonometry

Related Books

Related Books

k	[(θ+ 2kπ)/n] = [(0 + 2kπ)/6] = [(kπ)/3]	cosα+i sinα	Six Answers
0	0	1+0i	1
1	[(π)/3]	[1/2] + [(√3 )/2]i	[1/2] + [(√3 )/2]i
2	[(2π)/3]	− [1/2] + [(√3 )/2]i	− [1/2] + [(√3 )/2]i
3	π	− 1 + 0i	−1
4	[(4π)/3]	− [1/2] − [(√3 )/2]i	− [1/2] − [(√3 )/2]i

k	[(θ+ 2kπ)/n] = [(π+ 2kπ)/4]	r^{^[1/n]}(cosα+ isinα)	Four Answers
0	[(π)/4]	√2 ([(√2 )/2] + i[(√2 )/2])	1+i
1	[(3π)/4]	√2 ( − [(√2 )/2] + i[(√2 )/2])	−1+i
2	[(5π)/4]	√2 ( − [(√2 )/2] + i( − [(√2 )/2]) )	−1−i
3	[(7π)/4]	√2 ( [(√2 )/2] + i( − [(√2 )/2]) )	1−i

k	[(θ+ 2kπ)/n] = [(0 + 2kπ)/3]	2(cosα+ isinα)	Three Answers
0	0	2(1 + 0i)	2
1	[(2π)/3]	2( − [1/2] + i[(√3 )/2] )	− 1 + √3 i
2	[(4π)/3]	2( − [1/2] − i[(√3 )/2] )	− 1 − √3 i

k	[(θ+ 2kπ)/n] = [(π+ 2kπ)/3]	5(cosα+ isinα)	Three Answers
0	[(π)/3]	5( [1/2] + i[(√3 )/2] )	[5/2] + [(5√3 )/2]i
1	π	5( − 1 + i(0) )	−5
2	[(5π)/3]	5( [1/2] + i( − [(√3 )/2]) )	[5/2] − [(5√3 )/2]i

Educator

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DeMoivre's Theorem

Share this knowledge with your friends!

DeMoivre's Theorem

Convert the complex number w = − 1 + √3 i into polar form and then use DeMoivre's Theorem to calculate w5

− 16 − 16√{3 i}

Convert the complex number w = − 4√3 + 4i into polar form and then use DeMoivre's Theorem to calculate w6

− 262144

Convert the complex number w = − 2 + 2√{3 i} into polar form and then use DeMoivre's Theorem to calculate w7

− 8192 + 8192√{3i}

Convert the complex number w = − 3√2 − 3√{2i} into polar form and then use DeMoivre's Theorem to calculate w4

− 1296

Convert the complex number w = 4 + 4i into polar form and then use DeMoivre's Theorem to calculate w3

− 128 + 128i

Convert the complex number w = 2 − 2√{3 i} into polar form and then use DeMoivre's Theorem to calculate w4

− 128 + 128√{3i}

Find all sixth roots of 1

k [(θ+ 2kπ)/n] = [(0 + 2kπ)/6] = [(kπ)/3] cosα+i sinα Six Answers 0 0 1+0i 1 1 [(π)/3] [1/2] + [(√3 )/2]i [1/2] + [(√3 )/2]i 2 [(2π)/3] − [1/2] + [(√3 )/2]i − [1/2] + [(√3 )/2]i 3 π − 1 + 0i −1 4 [(4π)/3] − [1/2] − [(√3 )/2]i − [1/2] − [(√3 )/2]i

Find all fourth roots of - 4

Find all cube roots of 8

k [(θ+ 2kπ)/n] = [(0 + 2kπ)/3] 2(cosα+ isinα) Three Answers 0 0 2(1 + 0i) 2 1 [(2π)/3] 2( − [1/2] + i[(√3 )/2] ) − 1 + √3 i 2 [(4π)/3] 2( − [1/2] − i[(√3 )/2] ) − 1 − √3 i

Find all cube roots of -125

k [(θ+ 2kπ)/n] = [(π+ 2kπ)/3] 5(cosα+ isinα) Three Answers 0 [(π)/3] 5( [1/2] + i[(√3 )/2] ) [5/2] + [(5√3 )/2]i 1 π 5( − 1 + i(0) ) −5 2 [(5π)/3] 5( [1/2] + i( − [(√3 )/2]) ) [5/2] − [(5√3 )/2]i

Answer

DeMoivre's Theorem

Mathematics: Trigonometry

Related Books

Related Books

Convert the complex number w = − 1 + √3 i into polar form and then use DeMoivre's Theorem to calculate w⁵

Convert the complex number w = − 4√3 + 4i into polar form and then use DeMoivre's Theorem to calculate w⁶

Convert the complex number w = − 2 + 2√{3 i} into polar form and then use DeMoivre's Theorem to calculate w⁷

Convert the complex number w = − 3√2 − 3√{2i} into polar form and then use DeMoivre's Theorem to calculate w⁴

Convert the complex number w = 4 + 4i into polar form and then use DeMoivre's Theorem to calculate w³

Convert the complex number w = 2 − 2√{3 i} into polar form and then use DeMoivre's Theorem to calculate w⁴

k [(θ+ 2kπ)/n] = [(0 + 2kπ)/6] = [(kπ)/3] cosα+i sinα Six Answers

0 0 1+0i 1

1 [(π)/3] [1/2] + [(√3 )/2]i [1/2] + [(√3 )/2]i

2 [(2π)/3] − [1/2] + [(√3 )/2]i − [1/2] + [(√3 )/2]i

3 π − 1 + 0i −1

4 [(4π)/3] − [1/2] − [(√3 )/2]i − [1/2] − [(√3 )/2]i

k [(θ+ 2kπ)/n] = [(0 + 2kπ)/3] 2(cosα+ isinα) Three Answers

0 0 2(1 + 0i) 2

1 [(2π)/3] 2( − [1/2] + i[(√3 )/2] ) − 1 + √3 i

2 [(4π)/3] 2( − [1/2] − i[(√3 )/2] ) − 1 − √3 i

k [(θ+ 2kπ)/n] = [(π+ 2kπ)/3] 5(cosα+ isinα) Three Answers

0 [(π)/3] 5( [1/2] + i[(√3 )/2] ) [5/2] + [(5√3 )/2]i

1 π 5( − 1 + i(0) ) −5

2 [(5π)/3] 5( [1/2] + i( − [(√3 )/2]) ) [5/2] − [(5√3 )/2]i