Dr. William Murray teaches Educator's Trigonometry course covering Functions, Identities, Applications, and Complex/Polar Coordinates. Educated at UC Berkeley (Ph.D) and Georgetown (B.S.), Professor Murray returns to the class that made him fall in love with mathematics. He will guide you through topics such as the Pythagorean Identity, Inverse Trigonometric Functions, Law of Sines, Law of Cosines, Trigonometric Word Problems, Vectors, and end with Polar Coordinates. This course is essential to those having trouble with trigonometry in any setting as Will’s course meets or exceeds all state standards. Each video lecture is accompanied by several worked-out video examples and QuickNotes to help summarize each topic.

Table of Contents

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 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 
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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