Simplifying Complex Numbers, Pre Calculus

Simplifying Complex Numbers

I. Polar Coordinates and Complex Numbers : Lecture 4 | 38:44 min

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Simplifying Complex Numbers

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

Negative Sign Under Radical

Example: Simplifying Negative sign Under Radical

Like Terms

Example: Simplifying Like Terms

Rationalizing the Denominator

Distribution and FOIL

Complex Conjugates

Example: Complex Conjugates

Powers of i

Example: Find Powers of i

Extra Example 1: Simplify the Expression

Extra Example 2: Simplify the Expression

Extra Example 3: Simplify the Expression

Extra Example 4: Powers of i

I. Polar Coordinates and Complex Numbers
	Polar Coordinates	29:32
	Graphing Polar Coordinates	64:42
	Introduction to Complex Numbers	34:10
	Simplifying Complex Numbers	38:44
	Polar Forms of Complex Numbers	33:50
	Products and Quotients of Complex Numbers in Polar Form	26:12
	Powers and Roots of Complex Numbers and de Moivre's Formula	63:15
II. Mathematical Analysis
	Mathematical Induction	30:39
	Fundamental Theorem of Algebra	28:55
	Rational Functions	45:48
III. Trigonometry (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
IV. Trigonometry (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
V. Trigonometry (Applications)
	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
	DeMoivre's Theorem	57:37
VI. Vectors and Parametric Equations
	Geometric Vectors	40:10
	Algebraic Representation of Vectors	54:36
	Vectors in 3D Space	50:08
	Vectors and Parametric Equations	34:10
	Using Parametric Equations to Model Motion	31:13
VII. Linear Algebra (Matrices)
	Basic Matrix Concepts	11:34
	Matrix Operations	21:36
	Matrix Multiplication	29:36
	Determinants	33:13
	Cramer's Rule	28:25
	Identity and Inverse Matrices	22:25
	Gauss Jordan Method	58:48
VIII. Sequences and Series
	Arithmetic Sequences	21:16
	Arithmetic Series	21:36
	Geometric Sequences	23:03
	Geometric Series	22:43
	Infinite Geometric Series	18:32
	Recursion and Special Sequences	14:34
	Binomial Theorem	48:30
IX. Analytic Geometry (Conic Sections)
	Midpoint and Distance Formulas	32:42
	Parabolas	41:27
	Circles	21:03
	Ellipses	46:51
	Hyperbolas	38:15
	Conic Sections	18:43
	Solving Quadratic Systems	47:04
X. Probability
	Experiment, Outcomes, and Sample Space	14:54
	Calculating Probability	14:13
	Probability and Events	22:08
	Intersection of Events and the Multiplication Rule	19:58
	Union of Events and the Addition Rule	18:28
	Bayes' Rule	16:59
	Probability II	77:39

Mathematics: Pre Calculus

I. Polar Coordinates and Complex Numbers
	Polar Coordinates			29:32
		Intro		0:00
		Rectangular vs. Polar Coordinates		0:04
			Describing a Location of a Point on a Plane: Rectangular	0:07
			Describing a Location of a Point on a Plane: Polar Coordinate	3:04
		Coordinate Conversion		4:23
			Using SOHCAHTOA	4:24
		Rectangular to Polar		6:03
			Rectangular Coordinate to Polar Coordinates	6:04
		Polar to Rectangular		9:16
			Polar Coordinates to Rectangular Coordinates	9:19
		Equation Conversion		11:25
			Example: Circle with r = 2	11:40
		Linear Equation with Slope		14:57
			Example: Describing Line	14:58
		Vertical or Horizontal Line		18:16
				19:22
		Extra Example 1: Plot the Polar Coordinates		21:16
		Extra Example 2: plot the polar Coordinates		23:44
		Extra Example 3: Convert to Linear Equation		24:56
		Extra Example 4: Convert to Linear Equation		26:09
	Graphing Polar Coordinates			64:42
		Intro		0:00
		Six Types of Polar Graphs		0:07
			Line	0:31
			Circle	1:08
			Limacon	1:46
			Rose	2:16
			Lemniscate	2:50
			Spirals	3:15
		Step in Graphing		3:50
			Determine Type of Graph	4:27
			Use Information Given by a, b, and n	4:39
			Plot Critical Points	4:52
		Circle		6:43
			Circle: r = ncosθ	6:44
		Circle Example		8:10
			Example: r = 6sinθ	8:13
		Limacon		12:58
			Inner Loop	13:06
			Cardioid	20:18
		Limacon, cont.		26:00
			Dimpled	26:05
			Convex	31:40
		Rose		33:30
			r = a sin nθ	33:31
			Length of Petals = a	33:39
		Rose: Even and Odd		34:06
			Even n (2n Petals)	34:07
			Odd n (n Petals)	39:28
		Lemniscate		41:08
			'Bowtie'	41:09
			Length of Petals	41:25
			Number of Petals	41:29
		Lemniscate Example		41:46
			Example: r² = 9sin2θ	41:47
		Spiral		44:13
			Spiral: r = a + bθ	44:14
			Example: r = 2θ	44:34
		Other Curves		46:31
			Conchoid	47:20
			Hyperbolic Spiral	47:35
			Strophoid	47:39
			Logarithmic Spiral	48:00
		Extra Example 1: Identify Graph		49:20
		Extra Example 2: Identify Graph		51:51
		Extra Example 3: Identify Graph		54:22
		Extra Example 4: Identify Graph		58:42
	Introduction to Complex Numbers			34:10
		Intro		0:00
		Imaginary Numbers		0:06
			n-th Degree Polynomials	0:40
			Multiplicities and Complex Roots	1:04
			What is an Imaginary Number?	2:00
		Complex Numbers		7:27
			Complex Numbers: a + bi	7:55
			Electrical Engineering	9:00
		Operations with Complex Numbers		9:53
			Cartesian, Rectangular, or Algebraic Form	10:13
			1) Addition/ Subtraction	10:38
			2) Multiplication	12:30
			3) Rationalizing the Denominator (Using the Complex Conjugate)	15:05
		Powers of i		20:53
			i = √-1	20:58
		Extra Example 1: Add/ Subtract the Complex Numbers		25:03
		Extra Example 2: Multiply the Complex Numbers		26:09
		Extra Example 3: Powers of i		28:39
		Extra Example 4: Rationalize the Denominator Using Complex Conjugate		30:32
	Simplifying Complex Numbers			38:44
		Intro		0:00
		Negative Sign Under Radical		1:01
			Example: Simplifying Negative sign Under Radical	1:02
		Like Terms		3:54
			Example: Simplifying Like Terms	3:55
		Rationalizing the Denominator		5:36
			Example: Simplify 3/i	5:52
			Example: Simplify (3+i)/5i	7:47
		Distribution and FOIL		10:25
			Example: 6i(3 - 2i)	10:49
			Example: (2 + 4i)(2 - 6i)	12:02
		Complex Conjugates		13:54
			Example: Complex Conjugates	13:55
		Powers of i		19:17
			Example: Find Powers of i	19:18
		Extra Example 1: Simplify the Expression		24:28
		Extra Example 2: Simplify the Expression		28:03
		Extra Example 3: Simplify the Expression		32:42
		Extra Example 4: Powers of i		35:10
	Polar Forms of Complex Numbers			33:50
		Intro		0:00
		Complex Plane		0:25
			Definition and Example of Complex Plane	0:26
		Representing Complex Numbers		2:51
			Example: Representing Complex Numbers	2:55
		Absolute Value of a Complex Number		4:25
			Definition of the Absolute Value of a + bi	4:26
		Trigonometric or Polar Form of a Complex Number		6:38
			Trigonometric Form of the Complex Number z = a + bi	7:00
			Modulus of z	8:14
			Argument of z	8:27
		Euler's Formula		9:36
			Example: Euler's Formula	9:38
		Complex to Trigonometric		11:11
			Example: Write Complex Number in Trigonometric Form	11:12
		Complex to Standard Form		16:35
			Example: Write Complex Number in Standard Form	16:36
		Extra Example 1: Give the Polar Coordinates		21:26
		Extra Example 2: Give the Polar Coordinates		24:11
		Extra Example 3: Find the Rectangular Coordinates		26:03
		Extra Example 4: Find the Rectangular Coordinates		30:01
	Products and Quotients of Complex Numbers in Polar Form			26:12
		Intro		0:00
		Sum and Difference		0:14
			Example of Sum and Difference Formulas	0:16
		Multiplication		1:00
			Multiplication Formulas	1:24
		Multiplications Example		5:53
			Example: Find the Product of 2 Complex Numbers	5:54
		Division		8:25
			Methods for Division	8:26
			Euler's Equation	8:58
		Division Example		10:43
			Example: Find the Quotient of the Complex Numbers	10:45
		Extra Example 1: Find the Product		14:46
		Extra Example 2: Simplify		17:08
		Extra Example 3: Find the Quotient		20:18
		Extra Example 4: Find the Quotient		22:59
	Powers and Roots of Complex Numbers and de Moivre's Formula			63:15
		Intro		0:00
		Powers of Complex Numbers		0:19
			Methods of Finding Powers of Complex Numbers	0:47
		De Moivre's Formula		3:07
			De Moivre's Formula	3:40
			De Moivre's Theorem	3:51
			De Moivre's Formula Example	4:56
			Example: Using De Moivre's Theorem	4:57
		De Moivre's Formula Example		9:08
			Example: Using De Moivre's Theorem	9:09
		Roots of Complex Numbers		11:43
			Equation for Finding the n-th Roots of a Complex Number	11:44
		Origin of Formula		13:47
			Fundamental Theorem of Algebra	13:48
		Origin of Formula, cont.		17:37
			Origin of Formula	17:38
		n-th Root Example		22:53
			Example: Calculate the 6-th Roots	22:54
		Extra Example 1: De Moivre's Theorem		29:59
		Extra Example 2: Cube Roots of 8i		36:50
		Extra Example 3: Find the Four Fourth Roots of -16		44:43
		Extra Example 4: Find the 3 Cube Root of -2 + 2i		52:08
II. Mathematical Analysis
	Mathematical Induction			30:39
		Intro		0:00
		What is Induction?		0:07
			Example of Mathematical Induction	0:09
		Historical Background		1:36
			French Mathematician Pierre de Fermat	1:42
		Euler		2:55
			Leonhard Euler	2:57
		Principle of Mathematical Induction		4:05
			Example: Statement Involving Positive Integer	4:08
		Extra Example 1: Mathematical Induction		6:35
		Extra Example 2: Mathematical Induction		12:18
		Extra Example 3: Mathematical Induction		16:08
		Extra Example 4: Mathematical Induction		21:33
	Fundamental Theorem of Algebra			28:55
		Intro		0:00
		Fundamental Theorem of Algebra		0:08
			Fundamental Theorem of Algebra	0:09
			Example of the Fundamental Theorem of Algebra	1:34
		Rational Zero Test		2:40
			Rational Zero Test	2:41
			Example of the Rational Zero Test	3:30
		Binomial Factors		5:47
			Binomial Factors	5:48
			Example of the Binomial Factors	6:18
		Descarte Rule of Signs		9:41
			Descarte Rule of Signs	9:42
		Descarte Rule of Signs Example		11:23
			Example: Describing Polynomial	11:24
		Extra Example 1: Find Zeroes of a Polynomial Function		13:09
		Extra Example 2: Find all the Zeroes		17:37
		Extra Example 3: Find the Polynomial		21:28
		Extra Example 4: Descarte Rule of Signs		25:37
	Rational Functions			45:48
		Intro		0:00
		Graph of Rational Functions		0:06
			Rational Functions	0:07
		Vertical Asymptote		0:53
			Example: y=1/x	0:55
		Horizontal Asymptote		2:52
			Example: Horizontal Asymptote	2:53
		Horizontal Asymptote Cases		4:27
			1st Case	4:40
			2nd Case	7:15
			3rd Case	9:00
			Special Case	10:31
		Intercepts		13:06
			x-intercepts	13:15
			y-intercepts	14:04
		Discontinuities		15:29
			Example: Discontinuities	15:30
		Examples		17:08
			H(x)= 5/(x + 3)	17:12
		Extra Example 1: Graph and Identify the Function		22:02
		Extra Example 2: Graph and Identify the Function		29:04
		Extra Example 3: Graph and Identify the Function		34:27
		Extra Example 4: Graph and Identify the Function		41:29
III. Trigonometry (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 Menmonic: 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
IV. Trigonometry (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
V. Trigonometry (Applications)
	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
	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
VI. Vectors and Parametric Equations
	Geometric Vectors			40:10
		Intro		0:00
		Magnitude and Direction		0:07
			Describing Quantities	0:12
			William Rowan Hamilton	1:42
			James Maxwell	2:04
		Vector Representation		2:43
			Scalar	3:50
			2 Vectors may not be Collinear	4:30
		Algebraically or Geometrically		5:44
			Representing Vectors	5:45
		Adding and Subtracting Vectors		7:58
			Addition: u + v	8:15
			Subtraction: u - v	10:16
		Multiplying Vectors		12:57
			Scalar Multiplication: 2u + (1/3)v	12:58
		Unit Vectors		15:28
			Standard Vectors Along x and y Axis	15:29
		Extra Example 1: Sketch 3u - (1/2)v		17:58
		Extra Example 2: Sketch the Vector		20:31
		Extra Example 3: Resultant Velocity of a Plane		22:57
		Extra Example 4: Add the Scalar Multiples		35:33
	Algebraic Representation of Vectors			54:36
		Intro		0:00
		Component Form		0:08
			Component Form	0:29
			Using Standard Unit Vector i, j	1:31
		Vector Operations		3:21
			Addition	3:30
			Subtraction	4:24
			Scalar Multiplication	5:40
		Unit Vectors		6:17
			Example: Find the Unit Vector	6:36
			Example: Find the Angle	9:42
		Dot Product		12:27
			Example: Dot Product	12:28
		Determining Orthogonality		22:28
			Example: Orthogonal	22:29
			Parallel	26:07
		Projections		28:52
			Finding Vector Components (Projection)	28:54
		Extra Example 1: Calculate u - v		35:40
		Extra Example 2: Find the Unit Vector		36:41
		Extra Example 3: Find the Angle		39:32
		Extra Example 4: Find the Direction and Magnitude of the Resultant Force		42:25
	Vectors in 3D Space			50:08
		Intro		0:00
		Algebraic Vector Operations in 3D		0:07
			Addition	1:11
			Subtraction	3:10
			Scalar Multiplication	4:30
			Magnitude (Distance Formula)	4:55
		Algebraic Vector Operations in 3D		7:54
			Equation of a Sphere	8:07
			Midpoint Formula	8:56
			Dot Product	9:55
			Angle Between 2 Vectors	11:00
			Quadrants/Octants	14:24
		Cross Product		16:13
			Cross Product/ Determinant	16:17
		Vector and Planes		21:01
			Example: x +3y + 6z = 12	21:02
		Cartesian Equation		24:39
			Cartesian Equation	24:41
		Equation of the Plane		31:16
			Example: Find the Equation of the Plane	31:17
		Extra Example 1: Find u - v		33:34
		Extra Example 2: Are the Vectors Perpendicular?		35:20
		Extra Example 3: Find the Equation of the Plane		37:32
		Extra Example 4: Non-zero Vector, Equation of a Plane, and Area		40:16
	Vectors and Parametric Equations			34:10
		Intro		0:00
		Rectangular and Parametric Equations		0:08
			Introduction to Rectangular and Parametric Equations	0:18
		Converting Parametric to Rectangular		2:35
			Example: Converting Parametric to Rectangular	2:36
		Trigonometric Functions		7:12
			Example: Trigonometric Functions	7:13
		Converting Rectangular to Parametric		10:08
			Example: Converting Rectangular to Parametric	10:09
		Extra Example 1: Parametric Equations		11:43
		Extra Example 2: Rectangular Equations		15:05
		Extra Example 3: Rectangular Equation		18:24
		Extra Example 4: Describe the Cycloid		22:17
	Using Parametric Equations to Model Motion			31:13
		Intro		0:00
		Vector Equations		0:23
			Linear Motion	1:06
			Parametric Equations	3:56
		Vector Equations		5:17
			Parametric Equations	5:18
		Parametric Equations on Curves		7:41
			Example: Parametric Equations on Curves	7:42
		Lissajous Curve		12:17
			Examples: Lissajous Curve	12:18
		Extra Example 1: Vector Equation		16:55
		Extra Example 2: Velocity, Vector Equation, and Parametric Equations		18:34
		Extra Example 3: Velocity and Speed		22:27
		Extra Example 4: Hyperbola		22:52
VII. Linear Algebra (Matrices)
	Basic Matrix Concepts			11:34
		Intro		0:00
		What is a Matrix		0:26
			Brackets	0:46
			Designation	1:21
			Element	1:47
			Matrix Equations	1:59
		Dimensions		2:27
			Rows (m) and Columns (n)	2:37
			Examples: Dimensions	2:43
		Special Matrices		4:22
			Row Matrix	4:32
			Column Matrix	5:00
			Zero Matrix	6:00
		Equal Matrices		6:30
			Example: Corresponding Elements	6:36
		Example 1: Matrix Dimension		8:12
		Example 2: Matrix Dimension		9:03
		Example 3: Zero Matrix		9:38
		Example 4: Row and Column Matrix		10:26
	Matrix Operations			21:36
		Intro		0:00
		Matrix Addition		0:18
			Same Dimensions	0:25
			Example: Adding Matrices	1:04
		Matrix Subtraction		3:42
			Same Dimensions	3:48
			Example: Subtracting Matrices	4:04
		Scalar Multiplication		6:08
			Scalar Constant	6:24
			Example: Multiplying Matrices	6:32
		Properties of Matrix Operations		8:23
			Commutative Property	8:41
			Associative Property	9:08
			Distributive Property	9:44
		Example 1: Matrix Addition		10:24
		Example 2: Matrix Subtraction		11:58
		Example 3: Scalar Multiplication		14:23
		Example 4: Matrix Properties		16:09
	Matrix Multiplication			29:36
		Intro		0:00
		Dimension Requirement		0:17
			n = p	0:24
			Resulting Product Matrix (m x q)	1:21
			Example: Multiplication	1:54
		Matrix Multiplication		3:38
			Example: Matrix Multiplication	4:07
		Properties of Matrix Multiplication		10:46
			Associative Property	11:00
			Associative Property (Scalar)	11:28
			Distributive Property	12:06
			Distributive Property (Scalar)	12:30
		Example 1: Possible Matrices		13:31
		Example 2: Multiplying Matrices		17:08
		Example 3: Multiplying Matrices		20:41
		Example 4: Matrix Properties		24:41
	Determinants			33:13
		Intro		0:00
		What is a Determinant		0:13
			Square Matrices	0:23
			Vertical Bars	0:41
		Determinant of a 2x2 Matrix		1:21
			Second Order Determinant	1:37
			Formula	1:45
			Example: 2x2 Determinant	1:58
		Determinant of a 3x3 Matrix		2:50
			Expansion by Minors	3:08
			Third Order Determinant	3:19
			Expanding Row One	4:06
			Example: 3x3 Determinant	6:40
		Diagonal Method for 3x3 Matrices		13:24
			Example: Diagonal Method	13:36
		Example 1: Determinant of 2x2		18:59
		Example 2: Determinant of 3x3		20:03
		Example 3: Determinant of 3x3		25:35
		Example 4: Determinant of 3x3		29:22
	Cramer's Rule			28:25
		Intro		0:00
		System of Two Equations in Two Variables		0:16
			One Variable	0:50
			Determinant of Denominator	1:14
			Determinants of Numerators	2:23
			Example: System of Equations	3:34
		System of Three Equations in Three Variables		7:06
			Determinant of Denominator	7:17
			Determinants of Numerators	7:52
		Example 1: Two Equations		8:57
		Example 2: Two Equations		13:21
		Example 3: Three Equations		17:11
		Example 4: Three Equations		23:43
	Identity and Inverse Matrices			22:25
		Intro		0:00
		Identity Matrix		0:13
			Example: 2x2 Identity Matrix	0:30
			Example: 4x4 Identity Matrix	0:50
			Properties of Identity Matrices	1:24
			Example: Multiplying Identity Matrix	2:52
		Matrix Inverses		5:30
			Writing Matrix Inverse	6:07
		Inverse of a 2x2 Matrix		6:39
			Example: 2x2 Matrix	7:31
		Example 1: Inverse Matrix		10:18
		Example 2: Find the Inverse Matrix		13:04
		Example 3: Find the Inverse Matrix		17:53
		Example 4: Find the Inverse Matrix		20:44
	Gauss Jordan Method			58:48
		Intro		0:00
		Two Equations		0:13
			Example: Using Substitution/ Elimination	0:14
		Three Equations or Higher		2:39
			History	2:40
		Row Echelon Form and Back Substitution		3:58
			Row-Echelon Form	5:01
			Back Substitution	10:49
		Gaussian Elimination		12:32
			Steps in Gaussian Elimination	12:33
		Augmented Matrix		15:17
			Example: Augmented Matrix	15:18
			More Example: Augmented Matrix	17:14
		Gauss-Jordan Elimination		26:19
			Example: Gauss-Jordan Elimination	26:20
		Extra Example 1: Augmented Matrix		34:21
		Extra Example 2: Augmented Matrix and Row-Echelon Form		40:58
		Extra Example 3: Gaussian Elimination and Back Substitution		49:21
		Extra Example 4: Gauss-Jordan Elimination		52:34
VIII. Sequences and Series
	Arithmetic Sequences			21:16
		Intro		0:00
		Sequences		0:10
			General Form of Sequence	0:16
			Example: Finite/Infinite Sequences	0:33
		Arithmetic Sequences		0:28
			Common Difference	2:41
			Example: Arithmetic Sequence	2:50
		Formula for the nth Term		3:51
			Example: nth Term	4:32
		Equation for the nth Term		6:37
			Example: Using Formula	6:56
		Arithmetic Means		9:47
			Example: Arithmetic Means	10:16
		Example 1: nth Term		12:38
		Example 2: Arithmetic Means		13:49
		Example 3: Arithmetic Means		16:12
		Example 4: nth Term		18:26
	Arithmetic Series			21:36
		Intro		0:00
		What are Arithmetic Series?		0:11
			Common Difference	0:28
			Example: Arithmetic Sequence	0:43
			Example: Arithmetic Series	1:09
			Finite/Infinite Series	1:36
		Sum of Arithmetic Series		2:27
			Example: Sum	3:21
		Sigma Notation		5:53
			Index	6:14
			Example: Sigma Notation	7:14
		Example 1: First Term		9:00
		Example 2: Three Terms		10:52
		Example 3: Sum of Series		14:14
		Example 4: Sum of Series		18:13
	Geometric Sequences			23:03
		Intro		0:00
		Geometric Sequences		0:11
			Common Difference	0:38
			Common Ratio	1:08
			Example: Geometric Sequence	2:38
		nth Term of a Geometric Sequence		4:41
			Example: nth Term	4:56
		Geometric Means		6:51
			Example: Geometric Mean	7:09
		Example 1: 9th Term		12:04
		Example 2: Geometric Means		15:18
		Example 3: nth Term		18:32
		Example 4: Three Terms		20:59
	Geometric Series			22:43
		Intro		0:00
		What are Geometric Series?		0:11
			List of Numbers	0:24
			Example: Geometric Series	1:12
		Sum of Geometric Series		2:16
			Example: Sum of Geometric Series	2:41
		Sigma Notation		4:21
			Lower Index, Upper Index	4:38
			Example: Sigma Notation	4:57
		Another Sum Formula		6:08
			Example: n Unknown	6:28
		Specific Terms		7:41
			Sum Formula	7:56
			Example: Specific Term	8:11
		Example 1: Sum of Geometric Series		10:02
		Example 2: Sum of 8 Terms		14:15
		Example 3: Sum of Geometric Series		18:23
		Example 4: First Term		20:16
	Infinite Geometric Series			18:32
		Intro		0:00
		What are Infinite Geometric Series		0:10
			Example: Finite	0:29
			Example: Infinite	0:51
			Partial Sums	1:09
			Formula	1:37
		Sum of an Infinite Geometric Series		2:39
			Convergent Series	2:58
			Example: Sum of Convergent Series	3:28
		Sigma Notation		7:31
			Example: Sigma	8:17
		Repeating Decimals		8:42
			Example: Repeating Decimal	8:53
		Example 1: Sum of Infinite Geometric Series		12:15
		Example 2: Repeating Decimal		13:24
		Example 3: Sum of Infinite Geometric Series		15:14
		Example 4: Repeating Decimal		16:48
	Recursion and Special Sequences			14:34
		Intro		0:00
		Fibonacci Sequence		0:05
			Background of Fibonacci	0:23
			Recursive Formula	0:37
			Fibonacci Sequence	0:52
			Example: Recursive Formula	2:18
		Iteration		3:49
			Example: Iteration	4:30
		Example 1: Five Terms		7:08
		Example 2: Three Terms		9:00
		Example 3: Five Terms		10:38
		Example 4: Three Iterates		12:41
	Binomial Theorem			48:30
		Intro		0:00
		Pascal's Triangle		0:06
			Expand Binomial	0:13
			Pascal's Triangle	4:26
		Properties		6:52
			Example: Properties of Binomials	6:58
		Factorials		9:11
			Product	9:28
			Example: Factorial	9:45
		Binomial Theorem		11:08
			Example: Binomial Theorem	13:48
		Finding a Specific Term		18:36
			Example: Specific Term	19:26
		Example 1: Expand		24:39
		Example 2: Fourth Term		30:26
		Example 3: Five Terms		36:13
		Example 4: Three Iterates		45:07
IX. Analytic Geometry (Conic Sections)
	Midpoint and Distance Formulas			32:42
		Intro		0:00
		Midpoint Formula		0:15
			Example: Midpoint	0:30
		Distance Formula		2:30
			Example: Distance	2:52
		Example 1: Midpoint and Distance		4:58
		Example 2: Midpoint and Distance		8:07
		Example 3: Median Length		18:51
		Example 4: Perimeter and Area		23:36
	Parabolas			41:27
		Intro		0:00
		What is a Parabola?		0:20
			Definition of a Parabola	0:29
			Focus	0:59
			Directrix	1:15
			Axis of Symmetry	3:08
		Vertex		3:33
			Minimum or Maximum	3:44
		Standard Form		4:59
			Horizontal Parabolas	5:08
			Vertex Form	5:19
			Upward or Downward	5:41
			Example: Standard Form	6:06
		Graphing Parabolas		8:31
			Shifting	8:51
			Example: Completing the Square	9:22
			Symmetry and Translation	12:18
			Example: Graph Parabola	12:40
		Latus Rectum		17:13
			Length	18:15
			Example: Latus Rectum	18:35
		Horizontal Parabolas		18:57
			Not Functions	20:08
			Example: Horizontal Parabola	21:21
		Focus and Directrix		24:11
			Horizontal	24:48
		Example 1: Parabola Standard Form		25:12
		Example 2: Graph Parabola		30:00
		Example 3: Graph Parabola		33:13
		Example 4: Parabola Equation		37:28
	Circles			21:03
		Intro		0:00
		What are Circles?		0:08
			Example: Equidistant	0:17
			Radius	0:32
		Equation of a Circle		0:44
			Example: Standard Form	1:11
		Graphing Circles		1:47
			Example: Circle	1:56
		Center Not at Origin		3:07
			Example: Completing the Square	3:51
		Example 1: Equation of Circle		6:44
		Example 2: Center and Radius		11:51
		Example 3: Radius		15:08
		Example 4: Equation of Circle		16:57
	Ellipses			46:51
		Intro		0:00
		What Are Ellipses?		0:11
			Foci	0:23
		Properties of Ellipses		1:43
			Major Axis, Minor Axis	1:47
			Center	1:54
			Length of Major Axis and Minor Axis	3:21
		Standard Form		5:33
			Example: Standard Form of Ellipse	6:09
		Vertical Major Axis		9:14
			Example: Vertical Major Axis	9:46
		Graphing Ellipses		12:51
			Complete the Square and Symmetry	13:00
			Example: Graphing Ellipse	13:16
		Equation with Center at (h, k)		19:57
			Horizontal and Vertical	20:14
			Difference	20:27
			Example: Center at (h, k)	20:55
		Example 1: Equation of Ellipse		24:05
		Example 2: Equation of Ellipse		27:57
		Example 3: Equation of Ellipse		32:32
		Example 4: Graph Ellipse		38:27
	Hyperbolas			38:15
		Intro		0:00
		What are Hyperbolas?		0:12
			Two Branches	0:18
			Foci	0:38
		Properties		2:00
			Transverse Axis and Conjugate Axis	2:06
			Vertices	2:46
			Length of Transverse Axis	3:14
			Distance Between Foci	3:31
			Length of Conjugate Axis	3:38
		Standard Form		5:45
			Vertex Location	6:36
			Known Points	6:52
		Vertical Transverse Axis		7:26
			Vertex Location	7:50
		Asymptotes		8:36
			Vertex Location	8:56
			Rectangle	9:28
			Diagonals	10:29
		Graphing Hyperbolas		12:58
			Example: Hyperbola	13:16
		Equation with Center at (h, k)		16:32
			Example: Center at (h, k)	17:21
		Example 1: Equation of Hyperbola		19:20
		Example 2: Equation of Hyperbola		22:48
		Example 3: Graph Hyperbola		26:05
		Example 4: Equation of Hyperbola		36:29
	Conic Sections			18:43
		Intro		0:00
		Conic Sections		0:16
			Double Cone Sections	0:24
		Standard Form		1:27
			General Form	1:37
		Identify Conic Sections		2:16
			B = 0	2:50
			X and Y	3:22
		Identify Conic Sections, Cont.		4:46
			Parabola	5:17
			Circle	5:51
			Ellipse	6:31
			Hyperbola	7:10
		Example 1: Identify Conic Section		8:01
		Example 2: Identify Conic Section		11:03
		Example 3: Identify Conic Section		11:38
		Example 4: Identify Conic Section		14:50
	Solving Quadratic Systems			47:04
		Intro		0:00
		Linear Quadratic Systems		0:22
			Example: Linear Quadratic System	0:45
		Solutions		2:49
			Graphs of Possible Solutions	3:10
		Quadratic Quadratic System		4:10
			Example: Elimination	4:21
		Solutions		11:39
			Example: 0, 1, 2, 3, 4 Solutions	11:50
		Systems of Quadratic Inequalities		12:48
			Example: Quadratic Inequality	13:09
		Example 1: Solve Quadratic System		21:42
		Example 2: Solve Quadratic System		29:13
		Example 3: Solve Quadratic System		35:02
		Example 4: Solve Quadratic Inequality		40:29
X. Probability
	Experiment, Outcomes, and Sample Space			14:54
		Intro		0:00
		Basic Definitions		0:29
			Experiment	0:35
			Outcomes	0:55
			Sample Space	1:04
		Examples		1:34
			Roll a Die	1:39
			Flip a Coin	2:33
		Simple and Compound Events		3:30
			Event	3:43
			Simple Event	3:58
			Compound Event	4:27
		Example		5:14
		Extra Example 1		0:59
		Extra Example 2		4:21
	Calculating Probability			14:13
		Intro		0:00
		What is Probability		0:27
			Range	0:53
			Sum of Probabilities	1:26
			Example: Football Game	2:05
		Classical Probability		2:53
			Equally Likely Outcomes	3:05
			Example: Coin Flip	4:08
			Example: Die Roll	5:12
		Relative Frequency		6:44
			Example	7:22
		Subjective Probability		9:38
			Example	10:06
		Extra Example 1		1:04
		Extra Example 2		1:33
	Probability and Events			22:08
		Intro		0:00
		Mutually Exclusive Events		0:17
			Example: Coin Flip	0:27
			Example: Die Roll	3:03
		Independent Events		5:13
			Notation	3:31
			Example: Coin	6:01
		Independent Events, cont.		9:19
			Example: Coffee Drinkers	9:23
		Mutually Exclusive vs Independent		13:03
		Complementary Events		14:08
			Example: Coffee Drinkers	15:37
		Extra Example 1		1:16
		Extra Example 2		3:32
	Intersection of Events and the Multiplication Rule			19:58
		Intro		0:00
		Intersection of Events		0:08
			Venn Diagram	1:20
		Multiplication Rule		2:22
			Joint Probability	2:23
			Example	3:23
		Example		6:30
		Multiplication Rule for Independent Events		10:30
			Example	11:39
		Joint Probability of Mutually Exclusive Events		15:24
		Extra Example 1		1:24
		Extra Example 2		2:09
	Union of Events and the Addition Rule			18:28
		Intro		0:00
		Union of Events		0:06
			Venn Diagram	0:52
		Addition Rule		2:01
			Example: Coffee Drinkers	3:25
		Example		6:26
		Addition Rule for Mutually Exclusive Events		9:11
		Example		10:27
		Extra Example 1		2:41
		Extra Example 2		1:15
	Bayes' Rule			16:59
		Intro		0:00
		Partition of Events		0:07
			Venn Diagram	0:17
		Law of Total Probability		3:12
		Bayes' Rule		6:11
		Example		9:09
		Extra Example 1		4:07
	Probability II			77:39
		Intro		0:00
		What is Probability		0:06
			Foundation of Probability	0:31
			What is Probability?	1:34
		Terms		5:12
			Common Probability Terms	5:13
		Types of Probability Problems		13:41
			Probability of Either of 2 Events	13:44
		Types of Probability Problems		16:53
			Probability of Either of 2 Mutually Exclusive Events	17:08
		'At Least' Problems		18:29
			Example: 'At Least' Problem	18:30
		Likely Scenarios		24:16
			Cards, Dice, Flipping Coins, and Colored Marbles	24:17
			Example: 'and ' Scenario	24:54
			Example: 'or' Scenario	25:57
		Events Occurring Together		30:29
			Example: Conditional Probability	30:30
		Conditional Probability		37:10
			Problems Involving Conditional Probability	37:11
		Conditional Probability		44:48
			Example: Binomial Probability Theorem	44:57
		Combinations		51:00
			Examples: Combinations	51:14
		Expected Value		54:08
			Definition of Expected Value	54:09
			Example: Expected Value	54:27
		Extra Example 1: Conditional Probability		58:33
		Extra Example 2: Expected Value		63:39
		Extra Example 3: Marbles		72:52

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Simplifying Complex Numbers

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Simplifying Complex Numbers

Mathematics: Pre Calculus

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