SlideShare uma empresa Scribd logo
1 de 10
Baixar para ler offline
Sum and Difference Proofs
(continued ...)




             cosbmw by marymactavish
Proofs of the Sum and Difference Identities ...




                              Rotate ∠ROP so that OP
            O                 lies on the positive x-axis   O
Recall the Even and Odd Identities
 The sine and tangent functions are ODD functions.
 Cosine is an EVEN function.
  sin(-x) = -sin(x)        cos(-x) = cos(x)          tan(-x) = -tan(x)
Proofs of the Sum and Difference Identities ...




                              Rotate ∠ROP so that OP
            O                 lies on the positive x-axis   O
Pre-Cal 40S Slides March 24, 2008
Pre-Cal 40S Slides March 24, 2008
Pre-Cal 40S Slides March 24, 2008
Pre-Cal 40S Slides March 24, 2008
Pre-Cal 40S Slides March 24, 2008
Pre-Cal 40S Slides March 24, 2008

Mais conteúdo relacionado

Destaque

Faça sua empresa crescer e participe de um Pólo de Tecnologia
Faça sua empresa crescer e participe de um Pólo de TecnologiaFaça sua empresa crescer e participe de um Pólo de Tecnologia
Faça sua empresa crescer e participe de um Pólo de TecnologiaRodrigo Mesquita
 
Arquiteturas de software para computação ubiqua
Arquiteturas de software para computação ubiquaArquiteturas de software para computação ubiqua
Arquiteturas de software para computação ubiquaRubens Matos Junior
 
Relay For Life SL
Relay For Life SLRelay For Life SL
Relay For Life SLguest889bdd
 

Destaque (8)

Ppt98615
Ppt98615Ppt98615
Ppt98615
 
Biographies
BiographiesBiographies
Biographies
 
Faça sua empresa crescer e participe de um Pólo de Tecnologia
Faça sua empresa crescer e participe de um Pólo de TecnologiaFaça sua empresa crescer e participe de um Pólo de Tecnologia
Faça sua empresa crescer e participe de um Pólo de Tecnologia
 
Presentacion
PresentacionPresentacion
Presentacion
 
Business
BusinessBusiness
Business
 
Arquiteturas de software para computação ubiqua
Arquiteturas de software para computação ubiquaArquiteturas de software para computação ubiqua
Arquiteturas de software para computação ubiqua
 
Saludo de Pascua
Saludo de PascuaSaludo de Pascua
Saludo de Pascua
 
Relay For Life SL
Relay For Life SLRelay For Life SL
Relay For Life SL
 

Mais de Darren Kuropatwa

Providing Permission To Wonder v2.1
Providing Permission To Wonder v2.1Providing Permission To Wonder v2.1
Providing Permission To Wonder v2.1Darren Kuropatwa
 
Things That Suck About Digital Citizenship v1
Things That Suck About Digital Citizenship v1Things That Suck About Digital Citizenship v1
Things That Suck About Digital Citizenship v1Darren Kuropatwa
 
Digital Storytelling for Deeper Learning v1
Digital Storytelling for Deeper Learning v1Digital Storytelling for Deeper Learning v1
Digital Storytelling for Deeper Learning v1Darren Kuropatwa
 
Tales of Learning and the Gifts of Footprints v4.2
Tales of Learning and the Gifts of Footprints v4.2Tales of Learning and the Gifts of Footprints v4.2
Tales of Learning and the Gifts of Footprints v4.2Darren Kuropatwa
 
Making Student Thinking Visible v4
 Making Student Thinking Visible v4 Making Student Thinking Visible v4
Making Student Thinking Visible v4Darren Kuropatwa
 
Leveraging Digital for Deeper Learning
Leveraging Digital for Deeper LearningLeveraging Digital for Deeper Learning
Leveraging Digital for Deeper LearningDarren Kuropatwa
 
We Learn Through Stories at PRIZMAH17
We Learn Through Stories at PRIZMAH17We Learn Through Stories at PRIZMAH17
We Learn Through Stories at PRIZMAH17Darren Kuropatwa
 
Providing Permission to Wonder v3
Providing Permission to Wonder v3Providing Permission to Wonder v3
Providing Permission to Wonder v3Darren Kuropatwa
 
Making Student Thinking Visible v3.7
 Making Student Thinking Visible v3.7 Making Student Thinking Visible v3.7
Making Student Thinking Visible v3.7Darren Kuropatwa
 
Providing Permission to Wonder v2
Providing Permission to Wonder v2Providing Permission to Wonder v2
Providing Permission to Wonder v2Darren Kuropatwa
 
We Learn Through Stories v4 (master class)
We Learn Through Stories v4 (master class)We Learn Through Stories v4 (master class)
We Learn Through Stories v4 (master class)Darren Kuropatwa
 
Deep Learning Design: the middle ring
Deep Learning Design: the middle ringDeep Learning Design: the middle ring
Deep Learning Design: the middle ringDarren Kuropatwa
 

Mais de Darren Kuropatwa (20)

Behind Their Eyes v1
Behind Their Eyes v1Behind Their Eyes v1
Behind Their Eyes v1
 
Leading Change v1
Leading Change v1Leading Change v1
Leading Change v1
 
Providing Permission To Wonder v2.1
Providing Permission To Wonder v2.1Providing Permission To Wonder v2.1
Providing Permission To Wonder v2.1
 
Things That Suck About Digital Citizenship v1
Things That Suck About Digital Citizenship v1Things That Suck About Digital Citizenship v1
Things That Suck About Digital Citizenship v1
 
Digital Storytelling for Deeper Learning v1
Digital Storytelling for Deeper Learning v1Digital Storytelling for Deeper Learning v1
Digital Storytelling for Deeper Learning v1
 
Tales of Learning and the Gifts of Footprints v4.2
Tales of Learning and the Gifts of Footprints v4.2Tales of Learning and the Gifts of Footprints v4.2
Tales of Learning and the Gifts of Footprints v4.2
 
The Fourth Screen v4.2
The Fourth Screen v4.2The Fourth Screen v4.2
The Fourth Screen v4.2
 
Making Student Thinking Visible v4
 Making Student Thinking Visible v4 Making Student Thinking Visible v4
Making Student Thinking Visible v4
 
Learning is at BYTE 2017
Learning is at BYTE 2017Learning is at BYTE 2017
Learning is at BYTE 2017
 
Leveraging Digital for Deeper Learning
Leveraging Digital for Deeper LearningLeveraging Digital for Deeper Learning
Leveraging Digital for Deeper Learning
 
We Learn Through Stories at PRIZMAH17
We Learn Through Stories at PRIZMAH17We Learn Through Stories at PRIZMAH17
We Learn Through Stories at PRIZMAH17
 
Providing Permission to Wonder v3
Providing Permission to Wonder v3Providing Permission to Wonder v3
Providing Permission to Wonder v3
 
The Fourth Screen v4.1
The Fourth Screen v4.1The Fourth Screen v4.1
The Fourth Screen v4.1
 
Making Student Thinking Visible v3.7
 Making Student Thinking Visible v3.7 Making Student Thinking Visible v3.7
Making Student Thinking Visible v3.7
 
Learning is at AUHSD
Learning is at AUHSDLearning is at AUHSD
Learning is at AUHSD
 
Providing Permission to Wonder v2
Providing Permission to Wonder v2Providing Permission to Wonder v2
Providing Permission to Wonder v2
 
The Fourth Screen v4
The Fourth Screen v4The Fourth Screen v4
The Fourth Screen v4
 
Learning is at BLC16
Learning is at BLC16Learning is at BLC16
Learning is at BLC16
 
We Learn Through Stories v4 (master class)
We Learn Through Stories v4 (master class)We Learn Through Stories v4 (master class)
We Learn Through Stories v4 (master class)
 
Deep Learning Design: the middle ring
Deep Learning Design: the middle ringDeep Learning Design: the middle ring
Deep Learning Design: the middle ring
 

Último

ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...
ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...
ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...JojoEDelaCruz
 
ANG SEKTOR NG agrikultura.pptx QUARTER 4
ANG SEKTOR NG agrikultura.pptx QUARTER 4ANG SEKTOR NG agrikultura.pptx QUARTER 4
ANG SEKTOR NG agrikultura.pptx QUARTER 4MiaBumagat1
 
How to do quick user assign in kanban in Odoo 17 ERP
How to do quick user assign in kanban in Odoo 17 ERPHow to do quick user assign in kanban in Odoo 17 ERP
How to do quick user assign in kanban in Odoo 17 ERPCeline George
 
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptxMULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptxAnupkumar Sharma
 
Virtual-Orientation-on-the-Administration-of-NATG12-NATG6-and-ELLNA.pdf
Virtual-Orientation-on-the-Administration-of-NATG12-NATG6-and-ELLNA.pdfVirtual-Orientation-on-the-Administration-of-NATG12-NATG6-and-ELLNA.pdf
Virtual-Orientation-on-the-Administration-of-NATG12-NATG6-and-ELLNA.pdfErwinPantujan2
 
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...Nguyen Thanh Tu Collection
 
Textual Evidence in Reading and Writing of SHS
Textual Evidence in Reading and Writing of SHSTextual Evidence in Reading and Writing of SHS
Textual Evidence in Reading and Writing of SHSMae Pangan
 
4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptx4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptxmary850239
 
Oppenheimer Film Discussion for Philosophy and Film
Oppenheimer Film Discussion for Philosophy and FilmOppenheimer Film Discussion for Philosophy and Film
Oppenheimer Film Discussion for Philosophy and FilmStan Meyer
 
4.18.24 Movement Legacies, Reflection, and Review.pptx
4.18.24 Movement Legacies, Reflection, and Review.pptx4.18.24 Movement Legacies, Reflection, and Review.pptx
4.18.24 Movement Legacies, Reflection, and Review.pptxmary850239
 
Active Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdfActive Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdfPatidar M
 
Inclusivity Essentials_ Creating Accessible Websites for Nonprofits .pdf
Inclusivity Essentials_ Creating Accessible Websites for Nonprofits .pdfInclusivity Essentials_ Creating Accessible Websites for Nonprofits .pdf
Inclusivity Essentials_ Creating Accessible Websites for Nonprofits .pdfTechSoup
 
Field Attribute Index Feature in Odoo 17
Field Attribute Index Feature in Odoo 17Field Attribute Index Feature in Odoo 17
Field Attribute Index Feature in Odoo 17Celine George
 
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATIONTHEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATIONHumphrey A Beña
 
How to Add Barcode on PDF Report in Odoo 17
How to Add Barcode on PDF Report in Odoo 17How to Add Barcode on PDF Report in Odoo 17
How to Add Barcode on PDF Report in Odoo 17Celine George
 
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdf
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdfGrade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdf
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdfJemuel Francisco
 
Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17
Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17
Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17Celine George
 
ICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfVanessa Camilleri
 
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptx
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptxQ4-PPT-Music9_Lesson-1-Romantic-Opera.pptx
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptxlancelewisportillo
 

Último (20)

ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...
ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...
ENG 5 Q4 WEEk 1 DAY 1 Restate sentences heard in one’s own words. Use appropr...
 
ANG SEKTOR NG agrikultura.pptx QUARTER 4
ANG SEKTOR NG agrikultura.pptx QUARTER 4ANG SEKTOR NG agrikultura.pptx QUARTER 4
ANG SEKTOR NG agrikultura.pptx QUARTER 4
 
How to do quick user assign in kanban in Odoo 17 ERP
How to do quick user assign in kanban in Odoo 17 ERPHow to do quick user assign in kanban in Odoo 17 ERP
How to do quick user assign in kanban in Odoo 17 ERP
 
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptxMULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
MULTIDISCIPLINRY NATURE OF THE ENVIRONMENTAL STUDIES.pptx
 
Paradigm shift in nursing research by RS MEHTA
Paradigm shift in nursing research by RS MEHTAParadigm shift in nursing research by RS MEHTA
Paradigm shift in nursing research by RS MEHTA
 
Virtual-Orientation-on-the-Administration-of-NATG12-NATG6-and-ELLNA.pdf
Virtual-Orientation-on-the-Administration-of-NATG12-NATG6-and-ELLNA.pdfVirtual-Orientation-on-the-Administration-of-NATG12-NATG6-and-ELLNA.pdf
Virtual-Orientation-on-the-Administration-of-NATG12-NATG6-and-ELLNA.pdf
 
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...
HỌC TỐT TIẾNG ANH 11 THEO CHƯƠNG TRÌNH GLOBAL SUCCESS ĐÁP ÁN CHI TIẾT - CẢ NĂ...
 
Textual Evidence in Reading and Writing of SHS
Textual Evidence in Reading and Writing of SHSTextual Evidence in Reading and Writing of SHS
Textual Evidence in Reading and Writing of SHS
 
4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptx4.16.24 Poverty and Precarity--Desmond.pptx
4.16.24 Poverty and Precarity--Desmond.pptx
 
Oppenheimer Film Discussion for Philosophy and Film
Oppenheimer Film Discussion for Philosophy and FilmOppenheimer Film Discussion for Philosophy and Film
Oppenheimer Film Discussion for Philosophy and Film
 
4.18.24 Movement Legacies, Reflection, and Review.pptx
4.18.24 Movement Legacies, Reflection, and Review.pptx4.18.24 Movement Legacies, Reflection, and Review.pptx
4.18.24 Movement Legacies, Reflection, and Review.pptx
 
Active Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdfActive Learning Strategies (in short ALS).pdf
Active Learning Strategies (in short ALS).pdf
 
Inclusivity Essentials_ Creating Accessible Websites for Nonprofits .pdf
Inclusivity Essentials_ Creating Accessible Websites for Nonprofits .pdfInclusivity Essentials_ Creating Accessible Websites for Nonprofits .pdf
Inclusivity Essentials_ Creating Accessible Websites for Nonprofits .pdf
 
Field Attribute Index Feature in Odoo 17
Field Attribute Index Feature in Odoo 17Field Attribute Index Feature in Odoo 17
Field Attribute Index Feature in Odoo 17
 
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATIONTHEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
THEORIES OF ORGANIZATION-PUBLIC ADMINISTRATION
 
How to Add Barcode on PDF Report in Odoo 17
How to Add Barcode on PDF Report in Odoo 17How to Add Barcode on PDF Report in Odoo 17
How to Add Barcode on PDF Report in Odoo 17
 
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdf
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdfGrade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdf
Grade 9 Quarter 4 Dll Grade 9 Quarter 4 DLL.pdf
 
Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17
Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17
Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17
 
ICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdfICS2208 Lecture6 Notes for SL spaces.pdf
ICS2208 Lecture6 Notes for SL spaces.pdf
 
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptx
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptxQ4-PPT-Music9_Lesson-1-Romantic-Opera.pptx
Q4-PPT-Music9_Lesson-1-Romantic-Opera.pptx
 

Pre-Cal 40S Slides March 24, 2008

  • 1. Sum and Difference Proofs (continued ...) cosbmw by marymactavish
  • 2. Proofs of the Sum and Difference Identities ... Rotate ∠ROP so that OP O lies on the positive x-axis O
  • 3. Recall the Even and Odd Identities The sine and tangent functions are ODD functions. Cosine is an EVEN function. sin(-x) = -sin(x) cos(-x) = cos(x) tan(-x) = -tan(x)
  • 4. Proofs of the Sum and Difference Identities ... Rotate ∠ROP so that OP O lies on the positive x-axis O