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Professor Murray
Word Problems and Applications of Trigonometry
Slide Duration: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
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1 answer
Last reply by: Dr. William Murray
Tue Jan 7, 2014 11:55 AM
Post by Anwar Alasmari on January 1, 2014
Hello Doctor,
in the example III, why the result of the root of 350(190)(110)(50) is equals 100 times root of 35(19)(11)(5)?
3 answers
Last reply by: Dr. William Murray
Tue Aug 13, 2013 5:17 PM
Post by Taylor Wright on July 19, 2013
I thought that the Law of Sines doesn't work for AAA, because there would be infinite solutions.