Trigonometry (Part 2) Addition Formulae

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A2 Maths – Pure – Trigonometric Angle Addition Rules

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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7A Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Trigonometry & Modelling 1 • Addition Formulae • P2 Ex7A • 💡

Trigonometry & Modelling 2 • Creating 'tan' in expressions • P2 Ex7A • 💡

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

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(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 1-11✦

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 590a: Make simple use of angle sum formulae. (not yet active)
• 590b: Prove the angle sum formulae. (not yet active)
• 590c: Simplify more general expressions involving sin(A ± B) or cos(A ± B) (not yet active)
• 590: Exam Practice: Angle sum and double/half angle formulae for sin, cos and tan

Other Videos

The Addition Formulas

Courtesy of Maths Genie (Up to 3:29)

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• The Chalkface
  • Compound Angles Proof: Excel (if macros are blocked: right-click the file › Properties › tick Unblock)
• SRWhitehouse
  • Proof of Addition Formulae✦
  • Addition Formulae✦
  • Trigonometric addition formulae worksheet✦
• Ticktock Maths
  • Addition Formulae lesson (slides)

A2 Maths – Pure – Using Angle Addition Rules

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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7B Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Trigonometry & Modelling 3 • Uses of the Addition Formulae • P2 Ex7B • 💡

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths (From 2:36)

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 12-16✦

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 590d: Determine exact values for sin, cos and tan of more complex angles. (not yet active)
• 590m: Use the compound angle formulae leading to tan x = a
• 590: Exam Practice: Angle sum and double/half angle formulae for sin, cos and tan

Other Videos

The Addition Formulas

Courtesy of Maths Genie (From 3:29)

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• Ticktock Maths
  • Using the Angle Addition Formulae lesson (slides)

Worksheets

• Jethwa Maths Starter Activity (p3-4)
• kutasoftware
  • Sum and Difference Identities
  • Angle Sum/Difference Identities
• Solomon C3 Trigonometry D Questions✦ (Answers✦)
• MadAsMaths (T. Madas)
  • Line Coordinate Geometry Exam Questions Enrichment Q6 (difficulty 5 of 5)
  • Trigonometric Inverse Functions Basic Questions Q5–11, 13, 15–17, 19–20; Standard Questions Q4, 6, 8, 10, 16, 19; Hard Questions Q3, 5–7, 9–10, 14; Enrichment Questions Q1–2, 9–10 (difficulty 3–5 of 5)
  • Trigonometry Compound Angle Identities Q5–24
  • Trigonometry Exam Questions Basic Questions Q6, 8; Standard Questions Q1, 8–9, 15, 22–24, 31, 33, 35, 38, 42, 44, 49, 51, 54–55, 57, 59, 62–64, 74, 79, 82, 91, 98, 100, 110, 112, 114, 125, 134, 150–152, 163, 167, 172, 174, 179–180, 184–185, 189, 191; Hard Questions Q2, 10–11, 35, 38, 43, 47, 57, 60, 64, 70, 72, 77, 82, 84, 89, 96–97; Enrichment Questions Q3, 32, 37–38, 41, 74, 84–85, 102, 124–125, 127 (difficulty 2–5 of 5)
  • Advanced Mensuration Q8 (difficulty 5 of 5)
• Starting Points Maths
  • Addition and Double Angle Formulae (by Mr Cooke)

A2 Maths – Pure – Double Angle Formulae

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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7C Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Trigonometry & Modelling 4 • Double Angle Formulae • P2 Ex7C • 💡

Trigonometry & Modelling 5 • Using the Double Angle Formulae • P2 Ex7C • 💡

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 17-20✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 590e: Prove the double angle formulae. (not yet active)
• 590f: Make simple use of double angle formulae. (not yet active)
• 590h: Determine the value of sin(2x) or cos(2x) given sin(x) or cos(x) (not yet active)
• 590k: Use the double angle formulae backwards (half-angle formulae). (not yet active)
• 590: Exam Practice: Angle sum and double/half angle formulae for sin, cos and tan

Further Practice

• Practise Questions

Other Videos

The Double Angle Formulas

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• Andy Lutwyche
  • Codebreaker
• Math = Love (Sarah Carter)
  • Double Angle Joke worksheet
  • Double Angle Joke Worksheet
• Don Steward
  • sin 2x
• SRWhitehouse
  • Double and Half Angle Formulae✦
• Ticktock Maths
  • Double-Angle Formulae lesson (slides)

Worksheets

• Jethwa Maths Starter Activity (p1-2)
• kutasoftware
  • Multiple-Angle Identities
  • Double-/Half-Angle Identities
• MME
  • Double Angle Formulae: Questions | Answers
• MadAsMaths (T. Madas)
  • Trigonometry Exam Questions Basic Questions Q1, 4; Standard Questions Q27, 89, 111, 116, 118, 132, 147, 187, 190; Hard Questions Q3, 59, 93; Enrichment Questions Q2, 26, 33, 35, 53, 58 (difficulty 2–5 of 5)

A2 Maths – Pure – Proving Trig Identities

Proving Trig Identities with Double Angle Formulae

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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7F Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Trigonometry & Modelling 18 • Proving Identities using Double Angle • P2 Ex7F • 💡

Trigonometry & Modelling 19 • More Proving Identities • P2 Ex7F • 💡

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 33-36✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 590g: Simplify more general trigonometric expressions involving double angles. (not yet active)
• 590i: Prove identities involving sin, cos and tan that involve both single and double angles. (not yet active)
• 590l: Simplify expressions involving sin(3x) or cos(3x) (not yet active)
• 590n: Write sin A ± sin B or cos A ± cos B as a product of sin and cos (not yet active)
• 592c: Simplify expressions involving sec, textrmcosec and cot that also involve double angles.
• 592e: Simplify expressions involving sec², textrmcosec² and cot² that also involve double angles.
• 590: Exam Practice: Angle sum and double/half angle formulae for sin, cos and tan
• 592: Exam Practice: Constructing proofs involving sec, textrmcosec and cot

Further Practice

• Practise Questions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• Underground Mathematics
  • Equation or identity?✦
• Jo Morgan
  • Trig identity match✦ (by Jo Morgan)
• Ticktock Maths
  • Proving Trigonometric Identities lesson (slides)

Worksheets

• Jethwa Maths Starter Activity (p5-6), Worksheet (p1-3) & Topic Test (p3-5)
• martahidegkuti.com Trigonometric Identities 3
• MadAsMaths (T. Madas)
  • Trigonometric Inverse Functions Hard Questions Q2 (difficulty 4 of 5)
  • Trigonometry Double Angle Identities Q1–9
  • Trigonometry Exam Questions Basic Questions Q3, 11, 16, 19, 24–25, 28–29; Standard Questions Q5–6, 10, 13, 17, 25–26, 45, 52–53, 69, 75–77, 86, 88, 90, 95, 99, 101, 105, 107, 109, 120, 122, 126, 131, 133, 136, 142, 144, 146, 148, 153–154, 156, 164, 176, 182, 192; Hard Questions Q9, 12, 18–19, 22, 26, 28, 30–34, 41, 44, 49–50, 52, 55, 66, 81, 91, 99; Enrichment Questions Q42, 44, 57, 79, 87, 93, 101, 123 (difficulty 2–5 of 5)

A2 Maths – Pure – Solving Trigonometric Equations

Solving Trig Equations with Angle Addition Formula

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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7D Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Trigonometry & Modelling 6 • Solving Equations using Addition Formulae • P2 Ex7D • 💡

Trigonometry & Modelling 7 • Solving Equations using Double Angle • P2 Ex7D • 💡

Trigonometry & Modelling 8 • What happens when you 'cancel out' in trig equations? • P2 Ex7D • 💡

Trigonometry & Modelling 9 • Solving Equations involving Addition and Double • P2 Ex7D • 💡

Trigonometry & Modelling 10 • Exam Q including Double Angle • P2 Ex7D • 💡

Trigonometry & Modelling 11 • Tricky Q from the exercise • P2 Ex7D • 💡

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 21-25✦

Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 590j: Solve a trigonometric equation involving a mixture of single and double angles. (not yet active)
• 593f: Solve an equation involving sec, textrmcosec and cot and a mixture of single and double angles. (not yet active)
• 593g: Solve an equation involving sec, textrmcosec and cot that uses the angle sum formulae. (not yet active)
• 590: Exam Practice: Angle sum and double/half angle formulae for sin, cos and tan
• 593: Exam Practice: Solving trigonometric equations, involving sec, textrmcosec and cot, in an interval

Further Practice

• Practise Questions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• Ticktock Maths
  • Solving Trigonometric Equations lesson (slides)

Worksheets

• Jethwa Maths Starter Activity (p7-8)
• kutasoftware Equations and Multiple-Angle Identities (p1)
• Physics & Maths Tutor
  • Trigonometric Equations: Questions 2021–24 · Mark scheme · Model answers · Questions 2018–21 (with MS) · Model answers
• MadAsMaths (T. Madas)
  • Trigonometric Equations Q18–30
  • Trigonometric General Solutions Q17, 29–33 (difficulty 2–3 of 5)
  • Trigonometry Compound Angle Identities Q25–28
  • Trigonometry Double Angle Identities Q10–20
  • Trigonometry Exam Questions Basic Questions Q9–10, 21, 32; Standard Questions Q11, 16, 18–19, 21, 29, 37, 39, 70, 83, 103, 106, 113, 119, 121, 123, 129–130, 137, 143, 161, 165–166, 169–170; Hard Questions Q1, 5–8, 13–14, 23, 25, 36–37, 61, 63, 80, 83, 86, 94–95; Enrichment Questions Q29–30, 34, 39–40, 45–46, 49–51, 55, 59, 63, 71, 76, 122 (difficulty 3–5 of 5)

A2 Maths – Pure – Simplifying acos(x)+bsin(x)

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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7E Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Trigonometry & Modelling 12 • The Harmonic Identity • P2 Ex7E • 💡

Trigonometry & Modelling 13 • The Harmonic Identity, radians examples • P2 Ex7E • 💡

Trigonometry & Modelling 14 • Solving Equations using the Harmonic Identity • P2 Ex7E • 💡

Trigonometry & Modelling 15 • Maxima/Minima using the Harmonic Identity • P2 Ex7E • 💡

Trigonometry & Modelling 16 • More complex maxima/minima • P2 Ex7E • 💡

Trigonometry & Modelling 17 • Maxima/Minima involving Squaring • P2 Ex7E • 💡

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 26-32✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 591a: Write acos x+bsin x in the form Rcos(x+alpha)
• 591b: Sketch an equation written in R sin(x + alpha) form. (not yet active)
• 591c: Use a R sin(x + alpha) form to deduce the maximum or minimum value of the original expression.
• 591d: Calculate the value of x for which the maximum or minimum value of R sin(x + alpha) occurs.
• 591e: Use an R sin(x + alpha) form to deduce the maximum or minimum value of a more complex expression.
• 591f: Use an Rsin(x + alpha) form to solve an equation.
• 591g: Use an R sin(x + alpha) form to deduce the maximum or minimum value and corresponding value of x, where a variable substitution is required.
• 591: Exam Practice: Expressing a cos(x) + b sin(x) in the form R cos(x + alpha) or R sin(x+alpha)

Further Practice

• Practise Questions

Other Videos

The R Formulas

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• SRWhitehouse Alternative Form✦ (TES Account Required)
• Goal Free Problems
  • A-Level Trigonometry set Q14, Q26, Q27
• Ticktock Maths
  • Simplifying a cos x plus-minus b sin x lesson (slides)

Worksheets

• Jethwa Maths Starter Activity (p9-10)
• Solomon C3 Trigonometry F Questions✦ (Answers✦)
• MadAsMaths (T. Madas)
  • Functions Exam Questions Q121 (difficulty 4 of 5)
  • Trigonometry Exam Questions Basic Questions Q13, 20, 26; Standard Questions Q7, 32, 36, 41, 43, 71, 138, 145, 155, 157, 160, 168, 171, 173 (difficulty 3–4 of 5)
  • Trigonometry R Transformations (Harmonic Form) Q1–17

A2 Maths – Pure – Modelling with Trigonometric Functions

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

Download Blank Notes

Download Slides

7G Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Trigonometry & Modelling 20 • Modelling • P2 Ex7G • 💡

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 37-40✦

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 585e: Use a modelled context to determine the parameters in a sinusoidal model y = a sin (bx + c) + d, including understanding of phase shift. (not yet active)
• 585f: Use technology to determine the parameters in a sinusoidal model y = a sin (bx + c) + d (not yet active)
• 591h: Use an Rsin(x + alpha) form to determine the time required to reach a point in the trajectory in the original model.
• 585: Exam Practice: Properties of sinusoidal graphs (in radians) and sinusoidal modelling
• 591: Exam Practice: Expressing a cos(x) + b sin(x) in the form R cos(x + alpha) or R sin(x+alpha)

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• SRWhitehouse Maximums and Minimums in Alternative Form✦ (TES Account Required)
• Goal Free Problems
  • A-Level Trigonometry set Q10, Q29, Q30

Worksheets

• Jethwa Maths Starter Activity (p11-12), Topic Test (p11-14)
• Naikermaths Exam Qs (2005-2011) (Solutions), Exam Qs (2012-2017) (Solutions)
• Physics & Maths Tutor
  • Modelling with Trigonometric Functions: Questions 2021–24 · Mark scheme · Model answers · Questions 2018–21 (with MS) · Model answers
• MadAsMaths (T. Madas)
  • Trigonometry Exam Questions Standard Questions Q97, 127, 135, 159; Hard Questions Q27, 68, 73–74; Enrichment Questions Q80–81 (difficulty 4–5 of 5)
  • Trigonometry Introduction Exam Equations Q48–49 (difficulty 4 of 5)

External Whole Topic Past Paper Style Questions

• Dr Frost Topic Tests✦ (Teacher login required)
• Jethwa Maths Worksheet (p4-10), Topic Test (p6-10)
• Mathsaurus All Exam boards PPQs (Video Solutions Pending)
• Naikermaths Addition & Double Angle Formula Exam Qs (2005-2011) (Solutions), Exam Qs (2012-2017) (Solutions)
• PMT Past Papers by topic
  • AQA
    • C3 Trigonometry
  • Edexcel
    • C3 Trigonometric Equations,
    • Formulae,
    • Identities✦
    • Trigonometry
  • OCR
    • C3 Trigonometry
  • OCR MEI
    • C4 Identities & Functions – sec, cosec, cot 2 QP (MS)
    • C4 Equations 2 QP (MS), Equations 3 QP (MS)
    • C4 Compound & Double Angle Formulae 1 QP (MS), Compound & Double Angle Formulae 2 QP (MS)
  • Cheat Sheet
• Solomon C3 Trigonometry H Questions✦ (Answers✦)

Activities

• NRICH (student problems)
  • Trigonometry
• MEI
  • MEI teaching notes, misconceptions & prompts: Trigonometric identities✦
  • Three challenge questions✦
• Underground Mathematics
  • t for tan✦
• Andy Lutwyche
  • Erica's Errors On Trigonmetric Modelling

Worksheets

• MathedUp
  • Trig formulae: Questions
• Maths Genie
  • Trigonometry and Modelling: Questions · Solutions · Lesson · Topic page
• Jethwa Maths
  • Year 2 Trigonometry: Worksheets · Unit Test
• Mathsaurus (Kevin Olding)
  • Trigonometric identities (exam questions)
• MME
  • C4 Trigonometry: Questions | Answers
• SRWhitehouse
  • Mixed Trigonometry Questions✦
• OCR
  • Trigonometry Check-In Test✦
• MadAsMaths (T. Madas)
  • Trigonometric Identities (with Solutions) (difficulty 2–5 of 5)
  • Trigonometry Exam Questions Standard Questions Q81, 87, 104, 115, 140, 162, 178 (difficulty 4 of 5)