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Home » Mathematics » Trigonometry
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15:23

Trigonometry Online Course Dr. William Murray, Ph.D.

4.8
517 ratings • 32 reviews
  • Level Upper Intermediate
  • 23 Lessons (15hr : 23min)
  • 34,096 already enrolled!
  • Audio: English
  • English

Join Dr. William Murray in his Trigonometry online course which breaks down difficult-to-understand concepts with clear explanations and tons of example walkthroughs. Dr. Murray brings his 15+ years of math teaching experience to show you the importance of trigonometry in life as well as insights and strategies to do well in class.

Table of Contents

Section 1: 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 

Section 2: 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 

Section 3: 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 

Section 4: Complex Numbers and Polar Coordinates

  Polar Coordinates 1:07:35
   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 

Course Details:

Duration: 15 hours, 23 minutes

Number of Lessons: 23

This course is perfect for high school and college students taking Trigonometry. Almost every topic is covered and meets or exceeds all state standards. Each lesson also comes with in-depth study notes.

Additional Features:

  • Free Sample Lessons
  • Closed Captioning (CC)
  • Practice Questions
  • Downloadable Lecture Slides
  • Study Guides
  • Instructor Comments

Topics Include:

  • Sine & Cosine Values
  • Pythagorean Identity
  • Half-Angle & Double-Angle Formulas
  • Law of Sines & Cosines
  • Area of a Triangle
  • Polar Coordinates
  • Complex Numbers
  • DeMoivre’s Theorem

About Dr. Murray

Professor Murray received his Ph.D from UC Berkeley, B.S. from Georgetown University, and has been teaching in the university setting for 15+ years.

Student Testimonials:

"Hi Dr. Murray, I have never done this type of Trig before and found your explanation easy to understand." — Richard G.

“Thank you so much, Prof. Murray. It helped a lot. Your lecture on the special right triangles and how they relate to the unit circle helped a-LOT. I know my complete unit circle: radians, angles and coordinates too!” — Ivon N.

“May I commend and congratulate you on doing such an incredible job on explaining what I previously found such a difficult concept. Thanks again and I am finally understanding and enjoying Trig. You are a great teacher!!!!!” — Jonathan T.

"You're a great instructor. I've learned more from you in a couple days than this whole semester." — Safreeca L.

"You are a terrific instructor! Nobody ever explained me the concepts of trigonometry the way you do. I have made used of all of your lectures completely and want to thank you for it. I cannot think of being successful without these lectures and notes." — Varsha S.

Student Feedback

4.8

32 Reviews

44%
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By Peter KeJuly 21, 2016
Thank You!
By Tania TorresApril 26, 2016
Never mind, I now see the minor error in the video. Thank you for the wonderful course, Professor!
By David WuApril 28, 2015
Hi,Professor
At 10:48 you said -pi/4 is a positive term so we don't have to add pi to it, but it seems to me that -pi/4 is a negative term, wold you explain why?
Thank you
By Carroll FieldsNovember 1, 2014
I have another question. Why on the period portion of the question "B", are you writing for example, in Extra Example II:  2/3x, I thought it would be 2/3pi.

Thanks a lot for the lecture, it helped me very much in learning this concept.

Rusty
By Carroll FieldsNovember 1, 2014
Can you please explain again the math behind the vertical shift: -C/B.
How you  factored out B?



Thank You,
Rusty
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