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exact trigonometric values

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4 detailed 50-minute lessons with teaching scripts, worked examples, parent guides, and assessment criteria.

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Lesson Overview

Total Lessons: 4
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA

Learning Objectives

Prerequisites

Materials & Equipment

Lesson 1: Introduction: exact trigonometric values

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Write down everything you already know about exact trigonometric values. Then check against the key terms: Exact values, Why learn these?. Use a mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Exact values, Why learn these?.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: exact trigonometric values. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Mathematics because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
First Look

Start with the revision notes summary, then attempt: Write down the exact value of sin 60°.

Plenary (5 minutes)

Check Out

Your student states one thing they learned and one question they still have about exact trigonometric values.

Lesson 2: Core Concepts: exact trigonometric values

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

Quick recap: write 3 key points from Lesson 1 on exact trigonometric values. Check them against the notes below.

Main Content (35 minutes)

Exact values: are the precise values of trigonometric functions for special angles, written as fractions or surds rather than decimals.
Why learn these?: They appear frequently in exams and simplify calculations without a calculator.
ItemDetail
0°0
30°½
45°√2/2
60°√3/2
90°1
0°1
30°√3/2
45°√2/2

Practice (10 minutes)

Q: Write down the exact value of sin 60°.

Answer: √3/2

Plenary (5 minutes)

Explain Back

Your student teaches the key points back to you without looking. Fill any gaps immediately.

Lesson 3: Application: exact trigonometric values

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Recall the key terms: Exact values, Why learn these?. Define each in one sentence.

Main Content (35 minutes)

Parent/Teacher Guide: Let your student attempt each question alone first, then compare with the model answer. Award method marks for correct working even if the final answer is wrong.

Q1: Write down the exact value of sin 60°.

Answer: √3/2

Q2: Write down the exact value of tan 45°.

Answer: 1

Q3: Calculate sin 30° × cos 60°.

Answer: ½ × ½ = ¼

Q4: Find sin² 45° + cos² 45°.

Answer: (√2/2)² + (√2/2)² = ½ + ½ = 1

Q5: A right-angled triangle has angle 45° and hypotenuse 8 cm. Find the opposite side exactly.

Answer: sin 45° = opp/8, √2/2 = opp/8, opp = 8 × √2/2 = 4√2 cm

Plenary (5 minutes)

Error Review

Review any questions answered incorrectly. Identify whether the error was knowledge, method, or reading the question.

Lesson 4: Exam Practice: exact trigonometric values

Duration: 50 minutes

Starter Activity (5 minutes)

Command Words

Review what these command words require: state (one point), describe (say what happens), explain (say why), compare (both sides), evaluate (judgement).

Main Content (35 minutes)

Extended Answer

Extended question: Extended Answer 6 marks: An equilateral triangle has side length 2 cm. (a) Find the exact height using exact trig values. (b) Find the exact area. (c) A regular hexagon is made from 6 equilateral triangles. Find the exact area of the hexagon with side 2 cm. <div class="

(a) Height = 2 x sin(60) = 2 x root3/2 = root3 cm. (b) Area = 1/2 x 2 x root3 = root3 cm squared. (c) 6 equilateral triangles, each with area root3. Total area = 6 x root3 cm squared. Mark scheme: M1 sin(60), A1 root3, M1 area formula, A1 root3, M1 hexagon structure, A1 6 x root3

Exam Tips: Memorise all the exact values - they come up frequently | sin and cos at 45° are the same: √2/2 | Cos is sin read backwards | Remember sin²θ + cos²θ = 1 for any angle | Leave answers in exact form unless asked for decimals | Practise simplifying surds in your answers
Common Errors: Watch Out! 1. Wrong: Writing sin(45) = 0.707 instead of 1/root2 (or root2/2) Correct: Exact values must be left in surd form. sin(45) = 1/root2 = root2/2. Decimal approximations lose marks when exact values are requested. 2. Wrong: Confusing sin and cos values: saying sin(60) = 1/2 Correct: sin(60) = root3/2 and sin(30) = 1/2. Cos has the reverse pattern: cos(30) = root3/2 and cos(60) = 1/2. 3. Wrong: Rationalising incorrectly: writing 1/root2 as root2 instead of root2/2 Correct: 1/root2 = root2/2 (multiply top and bottom by root2). root2 is approximately 1.414, while 1/root2 is approximately 0.707.
AO3 - Reasoning & Interpretation: Reasoning and Interpretation A student calculates cos(30) as 0.866 on their calculator but the exam requires an exact answer. (a) Write cos(30) as an exact value in surd form. (b) Show that (root3/2) squared = 3/4. (c) A question asks for the exact area of a triangle with sides 10 cm, 10 cm and included angle 60 degrees. A student gives 43.3 cm squared. What should the exact answer be? Answers: (a) cos(30) = root3/2. (b) (root3/2) squared = 3/4. The root3 squared gives 3, and 2 squared gives 4. (c) Area = 1/2 x 10 x 10 x sin(60) = 50 x root3/2 = 25 x root3 cm squared. The decimal 43.3 is an approximation; the exact answer must be in surd form.
Stretch & Challenge (Grade 8-9):
  • Synoptic links: explain how exact trigonometric values connects to another Mathematics topic you have studied
  • Real-world: research one real-world use or example of exact trigonometric values
  • Critical: "What are the limitations of the models used in exact trigonometric values?"

Plenary (5 minutes)

Assessment Criteria
  • Got it: Confident explanation + correct worked examples
  • Getting there: Main points OK, needs support with detail
  • Not yet: Confused on key concepts - re-run Lesson 2

Homework & Consolidation

Recommended Resources

🎓 Smart Lesson (Guided)