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Professor Murray

Addition and Subtraction Formulas

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
- Half-Circle 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 Conversions-1
- Extra Example 2: Quadrants and Coterminal Angles-2

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 Triangle-1
- Extra Example 2: Graphing Sine and Cosine Functions-2

Sine and Cosine Values of Special Angles

33m 5s

- Intro0:00
- 45-45-90 Triangle and 30-60-90 Triangle0:08
- 45-45-90 Triangle0:21
- 30-60-90 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/Cosine-1
- Extra Example 2: All Angles and Quadrants-2

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 Shift-1
- Extra Example 2: Find Cosine Wave Given Attributes-2

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 Angles-1
- Extra Example 2: Tangent and Cotangent of Angles-2

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 Function-1
- Extra Example 2: Values of Secant and Cosecant-2

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 Arccosine-1
- Extra Example 2: Arcsin/Arccos/Arctan Values-2

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 Values-1
- Extra Example 2: Arcsin/Arccos/Arctan Values-2

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 Theorem-1
- Extra Example 2: Find Angle Given Cosine and Quadrant-2

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 Identity-1
- Extra Example 2: Given Sec Find Tan-2

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(A-B)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(A-B) and Cofunction Identities-1
- Extra Example 2: Convert to Radians and use Formulas-2

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 Angle-1
- Extra Example 2: Prove Trigonometric Identity using Double Angle-2

Half-Angle Formulas

43m 55s

- Intro0:00
- Main Formulas0:09
- Confusing Part0:34
- Example 1: Find Sine and Cosine of Angle using Half-Angle0:54
- Example 2: Prove Trigonometric Identity using Half-Angle11:51
- Example 3: Prove the Half-Angle Formula for Tangents18:39
- Extra Example 1: Find Sine and Cosine of Angle using Half-Angle-1
- Extra Example 2: Prove Trigonometric Identity using Half-Angle-2

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 Triangle-1
- Extra Example 2: Find Lengths of All Sides of Triangle-2

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 Completely-1
- Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely-2

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 Completely-1
- Extra Example 2: Length of Third Side and Area of Triangle-2

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 Angle-1
- Extra Example 2: Area of Triangle with Two Sides and One Angle-2

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 Height-1
- Extra Example 2: Roads to a Town-2

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 Vector-1
- Extra Example 2: Ship with a Current-2

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 Coordinates-1
- Extra Example 2: Graph the Polar Equation-2

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: Simplify-1
- Extra Example 2: All Complex Numbers Satisfying Equation-2

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 Forms-1
- Extra Example 2: Simplify Expression to Polar Form-2

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 Theorem-1
- Extra Example 2: Find Complex Fourth Roots-2

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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 Aug 17, 2018 7:03 PM

Post by John Stedge on August 9 at 04:57:43 PM

Extra Example 2 at 6:17, Bless you.

1 answer

Last reply by: Dr. William Murray

Sat Aug 13, 2016 10:54 AM

Post by tae Sin on August 12, 2016

I know this question is frivolous - I don't mind anyone answering this question if they know this is possible, but I used a calculator when I was bored. And I spammed sincostansincostan() with some random value - I'm pretty sure I used a real number that didn't create undefined value. And I actually got a value, so can you somehow explain how this is possible? if it does relate to the addition and subtraction formulas, please explain them as well?

2 answers

Last reply by: Dr. William Murray

Wed Apr 27, 2016 4:51 PM

Post by Tania Torres on April 26, 2016

Regarding Iris Kim's question, "At 12:14, you wrote that (cos(A-B)-1)^2 equals cos(A-B)^2-2cos(A-B)... shouldn't it be cos(A-B)^2-2cos(A-B)+1?" and your response, why is it not '+ 1'?

2 answers

Last reply by: Dr. William Murray

Wed Jul 1, 2015 8:52 AM

Post by Iris Kim on June 30, 2015

At 12:14, you wrote that (cos(A-B)-1)^2 equals cos(A-B)^2-2cos(A-B)... shouldn't it be cos(A-B)^2-2cos(A-B)+1?

1 answer

Last reply by: Dr. William Murray

Mon Nov 17, 2014 8:26 PM

Post by olga shevchuk on November 16, 2014

THERE WAS A MISTAKE. IT WAS WRITTEN 2SIN(3X)COS(X) WHEN IT SHOULD HAVE BEEN 2SIN(3X)COS(2X) STARTING @28:10

1 answer

Last reply by: Dr. William Murray

Tue Aug 5, 2014 3:49 PM

Post by Jamal Tischler on July 23, 2014

Very good lesson. I apreciate you derived the formulas ! It helped me.

3 answers

Last reply by: Dr. William Murray

Mon Jun 23, 2014 7:44 PM

Post by Jeffrey Tao on June 21, 2014

In your response to Manfred Berger's question, you stated how it is possible to use the Euler's formula, e^ix=cosx+isinx, to prove the identities,as a way that did not use calculus. But from what I've learned, the derivation of the formula e^ix=cosx+isinx comes from power series, so doesn't this method of proving the identities still use calculus?

1 answer

Last reply by: Dr. William Murray

Tue Dec 10, 2013 11:32 PM

Post by Monis Mirza on December 7, 2013

Write an equivalent expression for sin(2m)cos(n)+ cos(2m) sin(n)

3 answers

Last reply by: Dr. William Murray

Thu Jul 18, 2013 8:20 AM

Post by Manfred Berger on June 28, 2013

Are you going to prove any of the addition formulas in Calc 2?

1 answer

Last reply by: Dr. William Murray

Fri Aug 31, 2012 5:26 PM

Post by Su Jung Leem on August 2, 2012

I know it's a irrelevant question but i wasn't sure where to ask this question. Does anyeone know how to add y+2 over y squared - y -2 and one over 3y+3 ? I keep on getting different answers every time I trying to answer this question. please help!!!

1 answer

Last reply by: Dr. William Murray

Sun May 12, 2013 5:21 PM

Post by Nathan Thomas on January 8, 2012

He didn't include the tangent sum difference formulas which is very important and shouldn't be skipped over.

tan(a + b) =

(tan a + tan b) / (1 - (tan a)(tan b)

tan(a - b) =

(tan a - tan b) / (1 + (tan a)(tan b)

1 answer

Last reply by: Dr. William Murray

Sun May 12, 2013 5:19 PM

Post by Elina Bugar on August 23, 2011

how did he get the coordinates of angle a to be cosA,cosB

and for angle B (SinA, CosB)

3 answers

Last reply by: Dr. William Murray

Sun May 12, 2013 5:15 PM

Post by Marco Zendejo on June 22, 2011

Im kinda confuse in Example II.

How did Pie/12 turn into pie/4 - pie/6

If anyone could explain this I'll be grateful.

2 answers

Last reply by: Dr. William Murray

Sun May 12, 2013 5:11 PM

Post by Judith Gleco on June 11, 2011

Hi,

I was wandering if anyone is having any problems with the recording glitching, or stopping and going back to the begining of the lesson. Help so I know if it is my computer.

7 answers

Last reply by: Dr. William Murray

Sun May 12, 2013 5:07 PM

Post by Mark Mccraney on January 15, 2010

Lecture 3, ex 1: shouldn't the coords written in blue be A=(cosA, sinB) vs A=(cosA, cosB)