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Trigonometry Online Course Dr. William Murray, Ph.D.
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4.8

517 ratings • 32 reviews

Level Upper Intermediate
23 Lessons (15hr : 23min)
34,239 already enrolled!
4.8 517 Ratings & 32 Reviews
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
Course Details
About Dr. Murray

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

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

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%

By Peter Ke•July 21, 2016

Thank You!

By Tania Torres•April 26, 2016

Never mind, I now see the minor error in the video. Thank you for the wonderful course, Professor!

By David Wu•April 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 Fields•November 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 Fields•November 1, 2014

Can you please explain again the math behind the vertical shift: -C/B.
How you factored out B?

Thank You,
Rusty