Home » Mathematics » Trigonometry
No. of
Lectures
Duration
(hrs:min)
23
15:23

# Trigonometry Online CourseDr. William Murray, Ph.D. Facebook Twitter More

4.8
517 ratings • 32 reviews
• Level Upper Intermediate
• 23 Lessons (15hr : 23min)
• 34,229 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.

## Section 1: Trigonometric Functions

Angles 39:05
Intro 0:00
Degrees 0:22
Circle is 360 Degrees 0:48
Splitting a Circle 1:13
Circle is 2 Pi Radians 2:31
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
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

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.

• Free Sample Lessons
• Closed Captioning (CC)
• Practice Questions
• Study Guides

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

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.

Visit Dr. Murray’s page

#### Student Feedback

4.8

32 Reviews

44%
50%
3%
3%
0%
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
Hiï¼ŒProfessor
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

OR

### Start Learning Now

Our free lessons will get you started (Adobe Flash® required).
Get immediate access to our entire library.