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Table of Contents
I. Trigonometric Functions
Angles
39m 5s
 Intro0:00
 Degrees0:22
 Circle is 360 Degrees0:48
 Splitting a Circle1:13
 Radians2:08
 Circle is 2 Pi Radians2:31
 One Radian2:52
 HalfCircle and Right Angle4:00
 Converting Between Degrees and Radians6:24
 Formulas for Degrees and Radians6:52
 Coterminal, Complementary, Supplementary Angles7:23
 Coterminal Angles7:30
 Complementary Angles9:40
 Supplementary Angles10:08
 Example 1: Dividing a Circle10:38
 Example 2: Converting Between Degrees and Radians11:56
 Example 3: Quadrants and Coterminal Angles14:18
 Extra Example 1: Common Angle Conversions1
 Extra Example 2: Quadrants and Coterminal Angles2
Sine and Cosine Functions
43m 16s
 Intro0:00
 Sine and Cosine0:15
 Unit Circle0:22
 Coordinates on Unit Circle1:03
 Right Triangles1:52
 Adjacent, Opposite, Hypotenuse2:25
 Master Right Triangle Formula: SOHCAHTOA2:48
 Odd Functions, Even Functions4:40
 Example: Odd Function4:56
 Example: Even Function7:30
 Example 1: Sine and Cosine10:27
 Example 2: Graphing Sine and Cosine Functions14:39
 Example 3: Right Triangle21:40
 Example 4: Odd, Even, or Neither26:01
 Extra Example 1: Right Triangle1
 Extra Example 2: Graphing Sine and Cosine Functions2
Sine and Cosine Values of Special Angles
33m 5s
 Intro0:00
 454590 Triangle and 306090 Triangle0:08
 454590 Triangle0:21
 306090 Triangle2:06
 Mnemonic: All Students Take Calculus (ASTC)5:21
 Using the Unit Circle5:59
 New Angles6:21
 Other Quadrants9:43
 Mnemonic: All Students Take Calculus10:13
 Example 1: Convert, Quadrant, Sine/Cosine13:11
 Example 2: Convert, Quadrant, Sine/Cosine16:48
 Example 3: All Angles and Quadrants20:21
 Extra Example 1: Convert, Quadrant, Sine/Cosine1
 Extra Example 2: All Angles and Quadrants2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D
52m 3s
 Intro0:00
 Amplitude and Period of a Sine Wave0:38
 Sine Wave Graph0:58
 Amplitude: Distance from Middle to Peak1:18
 Peak: Distance from Peak to Peak2:41
 Phase Shift and Vertical Shift4:13
 Phase Shift: Distance Shifted Horizontally4:16
 Vertical Shift: Distance Shifted Vertically6:48
 Example 1: Amplitude/Period/Phase and Vertical Shift8:04
 Example 2: Amplitude/Period/Phase and Vertical Shift17:39
 Example 3: Find Sine Wave Given Attributes25:23
 Extra Example 1: Amplitude/Period/Phase and Vertical Shift1
 Extra Example 2: Find Cosine Wave Given Attributes2
Tangent and Cotangent Functions
36m 4s
 Intro0:00
 Tangent and Cotangent Definitions0:21
 Tangent Definition0:25
 Cotangent Definition0:47
 Master Formula: SOHCAHTOA1:01
 Mnemonic1:16
 Tangent and Cotangent Values2:29
 Remember Common Values of Sine and Cosine2:46
 90 Degrees Undefined4:36
 Slope and Menmonic: ASTC5:47
 Uses of Tangent5:54
 Example: Tangent of Angle is Slope6:09
 Sign of Tangent in Quadrants7:49
 Example 1: Graph Tangent and Cotangent Functions10:42
 Example 2: Tangent and Cotangent of Angles16:09
 Example 3: Odd, Even, or Neither18:56
 Extra Example 1: Tangent and Cotangent of Angles1
 Extra Example 2: Tangent and Cotangent of Angles2
Secant and Cosecant Functions
27m 18s
 Intro0:00
 Secant and Cosecant Definitions0:17
 Secant Definition0:18
 Cosecant Definition0:33
 Example 1: Graph Secant Function0:48
 Example 2: Values of Secant and Cosecant6:49
 Example 3: Odd, Even, or Neither12:49
 Extra Example 1: Graph of Cosecant Function1
 Extra Example 2: Values of Secant and Cosecant2
Inverse Trigonometric Functions
32m 58s
 Intro0:00
 Arcsine Function0:24
 Restrictions between 1 and 10:43
 Arcsine Notation1:26
 Arccosine Function3:07
 Restrictions between 1 and 13:36
 Cosine Notation3:53
 Arctangent Function4:30
 Between Pi/2 and Pi/24:44
 Tangent Notation5:02
 Example 1: Domain/Range/Graph of Arcsine5:45
 Example 2: Arcsin/Arccos/Arctan Values10:46
 Example 3: Domain/Range/Graph of Arctangent17:14
 Extra Example 1: Domain/Range/Graph of Arccosine1
 Extra Example 2: Arcsin/Arccos/Arctan Values2
Computations of Inverse Trigonometric Functions
31m 8s
 Intro0:00
 Inverse Trigonometric Function Domains and Ranges0:31
 Arcsine0:41
 Arccosine1:14
 Arctangent1:41
 Example 1: Arcsines of Common Values2:44
 Example 2: Odd, Even, or Neither5:57
 Example 3: Arccosines of Common Values12:24
 Extra Example 1: Arctangents of Common Values1
 Extra Example 2: Arcsin/Arccos/Arctan Values2
II. Trigonometric Identities
Pythagorean Identity
19m 11s
 Intro0:00
 Pythagorean Identity0:17
 Pythagorean Triangle0:27
 Pythagorean Identity0:45
 Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity1:14
 Example 2: Find Angle Given Cosine and Quadrant4:18
 Example 3: Verify Trigonometric Identity8:00
 Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem1
 Extra Example 2: Find Angle Given Cosine and Quadrant2
Identity Tan(squared)x+1=Sec(squared)x
23m 16s
 Intro0:00
 Main Formulas0:19
 Companion to Pythagorean Identity0:27
 For Cotangents and Cosecants0:52
 How to Remember0:58
 Example 1: Prove the Identity1:40
 Example 2: Given Tan Find Sec3:42
 Example 3: Prove the Identity7:45
 Extra Example 1: Prove the Identity1
 Extra Example 2: Given Sec Find Tan2
Addition and Subtraction Formulas
52m 52s
 Intro0:00
 Addition and Subtraction Formulas0:09
 How to Remember0:48
 Cofunction Identities1:31
 How to Remember Graphically1:44
 Where to Use Cofunction Identities2:52
 Example 1: Derive the Formula for cos(AB)3:08
 Example 2: Use Addition and Subtraction Formulas16:03
 Example 3: Use Addition and Subtraction Formulas to Prove Identity25:11
 Extra Example 1: Use cos(AB) and Cofunction Identities1
 Extra Example 2: Convert to Radians and use Formulas2
Double Angle Formulas
29m 5s
 Intro0:00
 Main Formula0:07
 How to Remember from Addition Formula0:18
 Two Other Forms1:35
 Example 1: Find Sine and Cosine of Angle using Double Angle3:16
 Example 2: Prove Trigonometric Identity using Double Angle9:37
 Example 3: Use Addition and Subtraction Formulas12:38
 Extra Example 1: Find Sine and Cosine of Angle using Double Angle1
 Extra Example 2: Prove Trigonometric Identity using Double Angle2
HalfAngle Formulas
43m 55s
 Intro0:00
 Main Formulas0:09
 Confusing Part0:34
 Example 1: Find Sine and Cosine of Angle using HalfAngle0:54
 Example 2: Prove Trigonometric Identity using HalfAngle11:51
 Example 3: Prove the HalfAngle Formula for Tangents18:39
 Extra Example 1: Find Sine and Cosine of Angle using HalfAngle1
 Extra Example 2: Prove Trigonometric Identity using HalfAngle2
III. Applications of Trigonometry
Trigonometry in Right Angles
25m 43s
 Intro0:00
 Master Formula for Right Angles0:11
 SOHCAHTOA0:15
 Only for Right Triangles1:26
 Example 1: Find All Angles in a Triangle2:19
 Example 2: Find Lengths of All Sides of Triangle7:39
 Example 3: Find All Angles in a Triangle11:00
 Extra Example 1: Find All Angles in a Triangle1
 Extra Example 2: Find Lengths of All Sides of Triangle2
Law of Sines
56m 40s
 Intro0:00
 Law of Sines Formula0:18
 SOHCAHTOA0:27
 Any Triangle0:59
 Graphical Representation1:25
 Solving Triangle Completely2:37
 When to Use Law of Sines2:55
 ASA, SAA, SSA, AAA2:59
 SAS, SSS for Law of Cosines7:11
 Example 1: How Many Triangles Satisfy Conditions, Solve Completely8:44
 Example 2: How Many Triangles Satisfy Conditions, Solve Completely15:30
 Example 3: How Many Triangles Satisfy Conditions, Solve Completely28:32
 Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely1
 Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely2
Law of Cosines
49m 5s
 Intro0:00
 Law of Cosines Formula0:23
 Graphical Representation0:34
 Relates Sides to Angles1:00
 Any Triangle1:20
 Generalization of Pythagorean Theorem1:32
 When to Use Law of Cosines2:26
 SAS, SSS2:30
 Heron's Formula4:49
 Semiperimeter S5:11
 Example 1: How Many Triangles Satisfy Conditions, Solve Completely5:53
 Example 2: How Many Triangles Satisfy Conditions, Solve Completely15:19
 Example 3: Find Area of a Triangle Given All Side Lengths26:33
 Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely1
 Extra Example 2: Length of Third Side and Area of Triangle2
Finding the Area of a Triangle
27m 37s
 Intro0:00
 Master Right Triangle Formula and Law of Cosines0:19
 SOHCAHTOA0:27
 Law of Cosines1:23
 Heron's Formula2:22
 Semiperimeter S2:37
 Example 1: Area of Triangle with Two Sides and One Angle3:12
 Example 2: Area of Triangle with Three Sides6:11
 Example 3: Area of Triangle with Three Sides, No Heron's Formula8:50
 Extra Example 1: Area of Triangle with Two Sides and One Angle1
 Extra Example 2: Area of Triangle with Two Sides and One Angle2
Word Problems and Applications of Trigonometry
34m 25s
 Intro0:00
 Formulas to Remember0:11
 SOHCAHTOA0:15
 Law of Sines0:55
 Law of Cosines1:48
 Heron's Formula2:46
 Example 1: Telephone Pole Height4:01
 Example 2: Bridge Length7:48
 Example 3: Area of Triangular Field14:20
 Extra Example 1: Kite Height1
 Extra Example 2: Roads to a Town2
Vectors
46m 42s
 Intro0:00
 Vector Formulas and Concepts0:12
 Vectors as Arrows0:28
 Magnitude0:38
 Direction0:50
 Drawing Vectors1:16
 Uses of Vectors: Velocity, Force1:37
 Vector Magnitude Formula3:15
 Vector Direction Formula3:28
 Vector Components6:27
 Example 1: Magnitude and Direction of Vector8:00
 Example 2: Force to a Box on a Ramp12:25
 Example 3: Plane with Wind18:30
 Extra Example 1: Components of a Vector1
 Extra Example 2: Ship with a Current2
IV. Complex Numbers and Polar Coordinates
Polar Coordinates
1h 7m 35s
 Intro0:00
 Polar Coordinates vs Rectangular/Cartesian Coordinates0:12
 Rectangular Coordinates, Cartesian Coordinates0:23
 Polar Coordinates0:59
 Converting Between Polar and Rectangular Coordinates2:06
 R2:16
 Theta2:48
 Example 1: Convert Rectangular to Polar Coordinates6:53
 Example 2: Convert Polar to Rectangular Coordinates17:28
 Example 3: Graph the Polar Equation28:00
 Extra Example 1: Convert Polar to Rectangular Coordinates1
 Extra Example 2: Graph the Polar Equation2
Complex Numbers
35m 59s
 Intro0:00
 Main Definition0:07
 Number i0:23
 Complex Number Form0:33
 Powers of Imaginary Number i1:00
 Repeating Pattern1:43
 Operations on Complex Numbers3:30
 Adding and Subtracting Complex Numbers3:39
 Multiplying Complex Numbers4:39
 FOIL Method5:06
 Conjugation6:29
 Dividing Complex Numbers7:34
 Conjugate of Denominator7:45
 Example 1: Solve For Complex Number z11:02
 Example 2: Expand and Simplify15:34
 Example 3: Simplify the Powers of i17:50
 Extra Example 1: Simplify1
 Extra Example 2: All Complex Numbers Satisfying Equation2
Polar Form of Complex Numbers
40m 43s
 Intro0:00
 Polar Coordinates0:49
 Rectangular Form0:52
 Polar Form1:25
 R and Theta1:51
 Polar Form Conversion2:27
 R and Theta2:35
 Optimal Values4:05
 Euler's Formula4:25
 Multiplying Two Complex Numbers in Polar Form6:10
 Multiply r's Together and Add Exponents6:32
 Example 1: Convert Rectangular to Polar Form7:17
 Example 2: Convert Polar to Rectangular Form13:49
 Example 3: Multiply Two Complex Numbers17:28
 Extra Example 1: Convert Between Rectangular and Polar Forms1
 Extra Example 2: Simplify Expression to Polar Form2
DeMoivre's Theorem
57m 37s
 Intro0:00
 Introduction to DeMoivre's Theorem0:10
 n nth Roots3:06
 DeMoivre's Theorem: Finding nth Roots3:52
 Relation to Unit Circle6:29
 One nth Root for Each Value of k7:11
 Example 1: Convert to Polar Form and Use DeMoivre's Theorem8:24
 Example 2: Find Complex Eighth Roots15:27
 Example 3: Find Complex Roots27:49
 Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem1
 Extra Example 2: Find Complex Fourth Roots2
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For more information, please see full course syllabus of Trigonometry
For more information, please see full course syllabus of Trigonometry
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1 answer
Last reply by: Dr. William Murray
Fri Jun 1, 2018 11:48 AM
Post by Kenneth Zhang on May 31 at 09:22:20 PM
How does the Law of Cosine turn into the Pythagorean Theorem in mathematical terms?
1 answer
Last reply by: Dr. William Murray
Wed May 30, 2018 9:41 AM
Post by Kenneth Zhang on May 28 at 07:26:59 PM
If one of the checks don't work, how might we solve for the problem then? Would we use imaginary numbers instead of proper formulas?
1 answer
Last reply by: Dr. William Murray
Fri Mar 20, 2015 6:29 PM
Post by Glenn O'Neill on March 19, 2015
in Example 2 you use the Law of Cosines to find each angle by relabeling all the angles to fit the formula... wouldn't it be a lot easier to label the triangle once, solve for angle C and then use the law of sines to find other other 2 angles?
2 answers
Last reply by: Dr. William Murray
Thu Jul 18, 2013 8:22 AM
Post by John Hunter on July 6, 2013
I hope i haven't missed the answer to this question somewhere in the lecture. You gave us the number of solutions for ASA, SAA, SSA, and AAA for the law of sines, but is there a shortcut to tell how many solutions a triangle will have when using the law of cosines with SAS and SSS? Many Thanks.
1 answer
Last reply by: Dr. William Murray
Thu May 30, 2013 4:01 PM
Post by Shyann Williams on March 29, 2011
Angle B = (angle AC)
1 answer
Last reply by: Dr. William Murray
Thu May 30, 2013 3:59 PM
Post by Mark Mccraney on January 18, 2010
There is a typo in Heron's formula as stated in Quicknotes: the fourth term, (sb), should be (sc). This will take in account for all three sides of the triangle, not 2 and side b being used twice.