Trigonometry Part 2 Identities & Equations

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AS Maths – Pure – Trigonometric Angles

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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10A Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Trigonometric Identities & Equations 1 • Recap • P1 Ex10A • 🚀

Trigonometric Identities & Equations 2 • Introducing the Unit Circle • P1 Ex10A • 🚀

Trigonometric Identities & Equations 3 • Exploring the Unit Circle • P1 Ex10A • 🚀

Trigonometric Identities & Equations 4 • Examples for the exercise • P1 Ex10A • 🚀

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✦

Extra Questions

• Dr Frost Questions (Short and no tan)

Key Skills

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

• 429d: Calculate the reference angle for an angle between 0 and 360 degrees. (not yet active)
• 429e: Calculate the reference angle for an angle above 360 degrees. (not yet active)
• 430a: Use the unit circle to find sin or cos of a multiple of 90^°
• 430b: Use the unit circle to find tan of a multiple of 90^°
• 430c: Use the unit circle to find sin or cos of a value in the second quadrant.
• 430d: Use the unit circle to find tan of a value in the second quadrant.
• 430e: Use the unit circle to find sin, cos or tan of a value in the third or fourth quadrant.
• 430f: Use the unit circle to find sin, cos or tan of angles above 360 degrees.
• 429: Exam Practice: Coterminal and reference angles
• 430: Exam Practice: Using the unit circle to find trigonometric values above 90^°

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Worksheets

• Dr Austin Maths
  • Trigonometric Ratios Decode the Joke (Word | PDF | Answers)
• pas1001 Symmetry in Trigonometric Graphs✦ (TES Account Required)

AS Maths – Pure – Exact Trigonometric Values

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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10B Slides (Courtesy of Owen134866)

Dr Frost Video

Exact Trigonometric Values

Whole Topic Video (GCSE)

Key Skill – Know exact trigonometric ratios.

K488: Know Exact Trigonometric Ratios.

Bicen Maths Live Lesson

Trigonometric Identities & Equations 5 • Where do exact values come from? • P1 Ex10B • 🚀

Trigonometric Identities & Equations 6 • Trigonometric Angles Laws • P1 Ex10B • 🚀

Trigonometric Identities & Equations 7 • Using exact values to find other ratios • P1 Ex10B • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

Dwnload Slides

• Dr Frost Slides 1-11✦

Exam Practice

• Exam Practice (GCSE)

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

• 412a: Know exact trigonometric ratios.
• 412b: Calculate the value of an expression using exact trigonometric ratios.
• 412c: Calculate a length in a single right-angled triangle using exact trigonometric ratios.
• 412d: Determine a length by using an exact trigonometric ratio combined with Pythagoras or further trigonometry.
• 412: Exam Practice: Exact trigonometric values

Practice Questions

• Practise Questions (GCSE)

Other Videos

The exact values for sin and cos 0, 30, 45, 60 and 90

Trigonometry Exact Values

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Phet Trig Tour
• Math = Love (Sarah Carter)
  • Quadrantal Angles Odd One Out
  • Evaluating Trig Functions Tarsia
  • Unit Circle activities
  • Fill in the Blank Unit Circle
  • Unit Circle Bingo
  • Unit Circle Projects
  • Exact Values Leap Frog Game
  • Unit Circle Paper Plate Activity
  • Trig Values Square Puzzle
  • Unit Circle Trig Ratios Notes
  • Unit Circle Notes
• NCTM Illuminations
  • Trigonometry Square
• Transum
  • Common Trig Ratios Degrees
• Interactive Maths
  • Trigonometric Exact Values: QQI | 10QQI | Count Down | Relay | Bingo

• Median Don Steward Sine Rule✦ some good exact values examples to use with prior knowledge

Worksheets

• Dr Austin Maths
  • Crack the Code (Word | PDF | Answers)
• Corbettmaths
  • Trigonometry: Exact Values Practice Questions & Solutions
  • Exact Trigonometric Values Worksheet✦ & Solutions
  • Exact Trig Values AQA Level 2 Practice Questions & Solutions
• kutasoftware Worksheets
  • Exact Trig Ratios of Important Angles *uses cast diagrams and extends into radians
• MME
  • Exact Values of Sin, Cos & Tan: Questions | Answers

AS Maths – Pure – Trigonometric Identities

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept. (From 8:56)

Tutorial Videos

A-Level Maths: E1-06 [Trigonometry: Finding Exact Values of sin(x), cos(x) & tan(x)]

A-Level Maths: E1-06 [Trigonometry: Finding Exact Values of sin(x), cos(x) & tan(x)] by TLMaths

How To Find The Exact Values of Trig Functions

How To Find The Exact Values of Trig Functions by The Organic Chemistry Tutor

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10C Part 1 Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Trigonometric Identities & Equations 8 • The Pythagorean Identity • P1 Ex10C • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths (using the Pythagorean Identity)

Extra Questions

• Dr Frost Questions (First 8/10 are Q’s submitted by WW and awaiting approval)

Key Skills

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

• 502a: Determine exact values of sin, cos and tan using another known trigonometric ratio, for acute angles.
• 502b: Determine exact values of sin, cos and tan using another known trigonometric ratio, for obtuse or reflex angles.
• 502c: Determine exact values of sin, cos and tan using another known trigonometric ratio in terms of an algebraic expression.
• 502: Exam Practice: Constructing proofs involving sin, cos and tan (excluding double angle formulae), and simplifying expressions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Worksheets

• Corbettmaths
  • Trigonometric Ratios AQA Level 2 Practice Questions & Solutions

AS Maths – Pure – Trigonometric Identities

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept. (up to 8:56)

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10C Part 2 Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video

Bicen Maths Live Lesson

Trigonometric Identities & Equations 9 • Proving Identities pt.1 • P1 Ex10C • 🚀

Trigonometric Identities & Equations 10 • Proving Identities pt.2 • P1 Ex10C • 🚀

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 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

• 502d: Prove that sin² x + cos² x equiv 1 (not yet active)
• 502e: Switch between sin² and cos² for expressions given in the form a sin² bx or a cos² bx
• 502f: Write an expression given in the form a sin² x + b cos² x in terms of just sin or cos
• 502g: Prove that tan x equiv sin x/cos x (not yet active)
• 502h: Simplify a trigonometric expression by splitting a fraction.
• 502i: Simplify a trigonometric expression by using the identity sin² x + cos² x equiv 1 and tan x = sin x/cos x
• 502j: Simplify a trigonometric expression involving tan² x or tan³ x
• 502k: Simplify an algebraic fraction in terms of sin and cos using quadratic factorisation.
• 502: Exam Practice: Constructing proofs involving sin, cos and tan (excluding double angle formulae), and simplifying expressions

Practice Questions

• Practise Questions
• Practise Questions (GCSE)

Other Videos

Trigonometric Identities 1

Courtesy of Maths Genie

Trigonometric Identities

Corbett Maths Trigonometric Identities

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities & Worksheets

• Dr Austin Maths
  • Match-Up (Word | PDF✦ | Answers✦)
  • Proof Practice Strips (Word | PDF✦ | Answers✦)
• Corbettmaths
  • Trigonometric Identities AQA Level 2 Practice Questions & Solutions
• Longman Cheshire Trigonometric Identities✦
• SRWhitehouse Trig Identities Worksheets✦ (TES Account Required)
• Math = Love (Sarah Carter)
  • Trig Identities Matching activity
  • Trig Identities Challenge

AS Maths – Pure – Simple Trigonometric Equations

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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10D Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video (Up to 8:54)

Bicen Maths Live Lesson

Trigonometric Identities & Equations 11 • Solving Simple Equations • P1 Ex10D • 🚀

Trigonometric Identities & Equations 12 • More examples: using your calc, Desmos • P1 Ex10D • 🚀

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 17-20✦

Exam Practice

• Exam Practice (GCSE)

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

• 503a: Solve a trigonometric equation given in the form sin x = k where x is in degrees.
• 503h: Determine the value of tan x given an equation in the form psin(x) = qcos(x) (not yet active)
• 434d: Use a graph to solve equations given in the form sin(x) = k or cos(x) = k in the range 0 to 360 degrees.
• 434e: Use a graph to solve equations given in the form tan(x) = k in the range 0 to 360 degrees.
• 434i: Solve equations given in the form sin(x) = sin(k)
• 434j: Solve equations given in the form sin(x) = -sin(k)
• 503: Exam Practice: Solving trigonometric equations, involving sin, cos and tan, in an interval in degrees
• 434: Exam Practice: Solving simple trigonometric equations, e.g. sin(x) = k, using a graph

Practice Questions

• Practise Questions (GCSE)

Other Videos

Solving Trigonometric Equations

Courtesy of Maths Genie (Up to 9:21 *Some Q’s use radians)

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Worksheets

• Dr Austin Maths
  • Using Graphs to Solve Trigonometric Equations Practice Grid ( Word | PDF | Answers)
• SRWhitehouse Trig Equations Worksheets✦ (TES Account Required)
• MathedUp
  • Trig Equations: Questions · Challenge · Answers
• Maths Genie
  • Solving Trigonometric Equations: Questions · Solutions · Lesson · Topic page
• Physics & Maths Tutor
  • Trigonometric Equations: Questions 2021–24 · Mark scheme · Model answers · Questions 2018–21 (with MS) · Model answers
• MadAsMaths (T. Madas)
  • Trigonometric Equations Introduction Q14–15
  • Trigonometry Introduction Exam Equations Q5–7 (difficulty 2 of 5)
• Starting Points Maths
  • Scaffolded Trig Equations

AS Maths – Pure – Harder Trigonometric Equations

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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10E Slides (Courtesy of Owen134866)

Dr Frost Video

KS5 – Solving Trigonometric Equations in an Interval

Whole Topic Video (From 8:54 to 14:07)

Bicen Maths Live Lesson

Trigonometric Identities & Equations 13 • Harder Equations with Linear Arguments • P1 Ex10E • 🚀

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 21-24✦

Exam Practice

• Exam Practice (Not Last Question)

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

• 503b: Solve a trigonometric equation given in the form sin(ax) = k where x is in degrees.
• 503c: State the number of solutions to a trigonometric equation given in the form sin(ax) = k or cos(ax) = k or tan(ax) = k for a given range.
• 503d: Solve a trigonometric equation given in the form sin (ax+b) = k where x is in degrees and k is positive.
• 503e: Solve a trigonometric equation given in the form sin (ax+b) = k where x is in degrees and k is negative.
• 503g: Solve a trigonometric equation in terms of sin(ax) where the equation involves a fraction.
• 503i: Solve a trigonometric equation given in the form psin (ax+b) = qcos (ax+b)
• 503m: Use the solutions from a trigonometric equation to deduce the solutions to a related equation. (not yet active)
• 503n: Determine the constants in y = sin(ax + b) given its roots on the graph. (not yet active)
• 503o: Solve a trigonometric equation from a modelled scenario.
• 503: Exam Practice: Solving trigonometric equations, involving sin, cos and tan, in an interval in degrees

Practice Questions

• Practise Questions

Other Videos

Solving Trigonometric Equations

Courtesy of Maths Genie (From 9:21 *Some Q’s use radians)

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Worksheets

• Jethwa Maths Starter Activity 1 (p3-4)
• Jethwa Maths Starter Activity 2 (p1-2 & 7-8) Q3 introduces Quadratic Concept – good to use at start of that lesson, Final Q on Modelling
• MadAsMaths (T. Madas)
  • Trigonometric Equations Introduction Q1–6, 8
  • Trigonometry Introduction Exam Equations Q1–3, 8–9, 11–14, 20, 23–24, 32, 35, 64, 68 (difficulty 2–4 of 5)

AS Maths – Pure – Trigonometric Equations and Identities

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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Download Slides

10F Slides (Courtesy of Owen134866)

Dr Frost Video

KS5 – Solving Trigonometric Equations in an Interval

Whole Topic Video (From 14:07)

Bicen Maths Live Lesson

Trigonometric Identities & Equations 14 • Quadratics in sin/cos/tan • P1 Ex10F • 🚀

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 25-28✦

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

• 503f: Solve trigonometric equations given in the form sin² x=a where x is in degrees.
• 503j: Solve a quadratic equation involving a single trigonometric function.
• 503k: Solve a trigonometric equation involving a mixture of sin and cos where one is squared.
• 503l: Solve a trigonometric equation involving tan and either sin or cos (not yet active)
• 503: Exam Practice: Solving trigonometric equations, involving sin, cos and tan, in an interval in degrees

Other Videos

Solving Quadratic Trigonometric Equations

Courtesy of Maths Genie (Some Q’s use radians)

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• MEI Solving Trigonometric Equations✦

Worksheets

• Dr Austin Maths
  • Solving Practice Strips (Word | PDF | Answers)
  • Solving Name the Film (Word | PDF | Answers)
• Jethwa Maths Starter Activity 2 (p3-6)
• MadAsMaths (T. Madas)
  • Trigonometric Equations Introduction Q18–28, 33–34, 37
  • Trigonometry Introduction Exam Equations Q17, 19, 22, 25, 27, 30, 33–34, 37–39, 44–45, 47, 51, 53–54, 57–58, 62, 66, 69–73 (difficulty 3–5 of 5)

Whole Topic Videos

Trigonometric Identities & Equations 15 • Exam Questions • P1 Ex10 Mixed • 🚀

Bicen Maths Exam Questions

External Whole Topic Past Paper Style Questions

• Dr Frost Topic Test✦ (Teacher Log in Required)
• Mathsaurus Mixed Exam Board PPQs
• Naiker Maths Exam Qs (2005-2011)✦ & MS✦
• PMT (any old spec questions may include A2 material – Radians & Sectors)
  • Summary Trigonometric Equations PPQs
  • Summary Trigonometric Identities PPQs
  • Edexcel C2 Set 1 Trigonometric identities PPQs (Old Spec)
  • Edexcel C2 Set 1 Trigonometric equations PPQs (Old Spec)
  • Edexcel C2 Set 2 Trigonometry PPQs (Old Spec)
  • OCR Set 1 C2 Trigonometry PPQs & MS (Old Spec)
  • OCR Set 2 C2 Trigonometry PPQs (Old Spec)
  • OCR MEI C2 Trigonometry – Basic Graphs PPQs & MS (Old Spec) – Some Equation Solving & Identities
  • OCR MEI C2 Trigonometry – Basic Identities 1 PPQs & MS (Old Spec)
  • OCR MEI C2 Trigonometry – Basic Identities 2 PPQs & MS (Old Spec)
  • Summary Cheat Sheet
• Pret Homework (@rachelwalters70)
  • Trig Identities and Equations (@rachelwalters70, pptx)

Worksheets

• Mathshelper (Jon Stone)
  • Trigonometric Equations✦
  • Trigonometry 2
• Maths Genie
  • Trigonometric Identities and Equations: Questions · Solutions · Lesson · Topic page
• Jethwa Maths
  • Year 1 Trigonometry: Worksheets · Unit Test
• Dr Oliver Mathematics
  • Trigonometry exam questions✦ (solutions✦)
• Dr Austin Maths
  • Exact Trigonometric Values Sort It Out (Word | PDF | Answers)
  • Solving Trigonometric with Exact Values Crack the Code (Word | PDF | Answers)
  • Solving Simple Trigonometric Fill in the Blanks (Word | PDF | Answers)
  • Solving Trigonometric with Multiple Angles Fill in the Blanks (Word | PDF | Answers)
  • Solving Quadratic Trigonometric Fill in the Blanks (Word | PDF | Answers)

Activities

• NRICH (student problems)
  • Trigonometry
• MEI
  • MEI teaching notes, misconceptions & prompts: Trigonometry✦
• Andy Lutwyche
  • Erica’s Errors On Trigonometric