G21: Exact Trigonometric Values
Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60°, 90°; know exact value of tan θ for θ = 0°, 30°, 45°, 60°
📋 Key Concepts
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.
📝 Sin Values
| Angle | sin θ |
|---|---|
| 0° | 0 |
| 30° | ½ |
| 45° | √2/2 |
| 60° | √3/2 |
| 90° | 1 |
Pattern: sin goes from 0 to 1 in a predictable way: 0, ½, √2/2, √3/2, 1
📝 Cos Values
| Angle | cos θ |
|---|---|
| 0° | 1 |
| 30° | √3/2 |
| 45° | √2/2 |
| 60° | ½ |
| 90° | 0 |
Pattern: cos is sin "backwards": 1, √3/2, √2/2, ½, 0
Note: sin 30° = cos 60°, sin 45° = cos 45°, sin 60° = cos 30°
📝 Tan Values
| Angle | tan θ |
|---|---|
| 0° | 0 |
| 30° | √3/3 or 1/√3 |
| 45° | 1 |
| 60° | √3 |
Remember: tan θ = sin θ / cos θ
Note: tan 90° is undefined (division by zero).
📝 Memory Tips
Sin table: Write as fractions with denominator 2:
- sin 0° = √0/2 = 0
- sin 30° = √1/2 = ½
- sin 45° = √2/2
- sin 60° = √3/2
- sin 90° = √4/2 = 1
Cos is sin backwards: Just reverse the order!
📝 Using Exact Values in Calculations
Example 1
Find the exact value of 2 sin 30° + cos 60°.
Solution:
sin 30° = ½, cos 60° = ½
2 sin 30° + cos 60° = 2(½) + ½ = 1 + ½ = 3/2
Example 2
Find the exact value of sin 45° × cos 45°.
Solution:
sin 45° = √2/2, cos 45° = √2/2
sin 45° × cos 45° = (√2/2)² = 2/4 = ½
Example 3
Find the exact value of tan² 45° + sin 60°.
Solution:
tan 45° = 1, sin 60° = √3/2
tan² 45° + sin 60° = 1² + √3/2 = 1 + √3/2 = (2 + √3)/2
📝 Finding Sides with Exact Values
Example 4
In a right-angled triangle, angle = 30°, hypotenuse = 10 cm. Find the opposite side exactly.
Solution:
sin 30° = opp/hyp
½ = opp/10
opp = 10 × ½ = 5 cm
Example 5
In a right-angled triangle, angle = 60°, adjacent = 6 cm. Find the hypotenuse exactly.
Solution:
cos 60° = adj/hyp
½ = 6/hyp
hyp = 6 × 2 = 12 cm
🧮 Worked Examples — Exam Style
Exact trig values are 'write down' marks — but only from the table or the two special triangles. Never reach for a calculator: √3/2 typed as 0.866… scores zero.
Write down the exact value of cos 30°
From the table: cos 30° = √3/2.
Where it comes from: half an equilateral triangle of side 2 gives sides 1, 2, √3 — cos 30° = adjacent/hypotenuse = √3/2.
B1 for √3/2. Writing 0.866 scores 0 — the question says exact.
Write down the exact value of tan 30°
Same half-equilateral triangle: tan 30° = opposite/adjacent = 1/√3.
Rationalised: 1/√3 = √3/3 (multiply top and bottom by √3).
B1 for 1/√3 or √3/3 — both forms are accepted. 0.577 is not.
Exam wording to expect: "Write down the exact value of …" — the word exact bans decimals. Memorise the table for 0°, 30°, 45°, 60°, 90°.
❓ Practice Questions
Q1: Write down the exact value of sin 60°.
Q2: Write down the exact value of tan 45°.
Q3: Calculate sin 30° × cos 60°.
Q4: Find sin² 45° + cos² 45°.
Q5: A right-angled triangle has angle 45° and hypotenuse 8 cm. Find the opposite side exactly.
Q6: Write down the exact value of cos 0°
Q7: Write down the exact value of tan 60°
✅ Answers
- √3/2
- 1
- ½ × ½ = ¼
- (√2/2)² + (√2/2)² = ½ + ½ = 1
- sin 45° = opp/8, √2/2 = opp/8, opp = 8 × √2/2 = 4√2 cm
🧮 Worked Examples — Harder Exam Style
A right-angled triangle has a 30° angle and a hypotenuse of 10 cm. Write down the exact length of the side opposite the 30° angle.
sin 30° = opposite ÷ hypotenuse
sin 30° = 1/2, so opposite = 10 × 1/2 = 5 cm
M1 for using sin 30° = 1/2 (exact form), A1 for 5. Writing 0.5 instead of 1/2 still scores here, but quoting 1/2 keeps 'exact' habits intact.
📋 Model Answers — Full Working
Every mark in GCSE maths comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. sin 60° = √3/2 (from the table, or the half-equilateral triangle)
Q2. tan 45° = 1 — the isosceles right triangle has equal legs
Q3. sin 30° = 1/2 and cos 60° = 1/2, so 1/2 × 1/2 = 1/4
Q4. (√2/2)² + (√2/2)² = 1/2 + 1/2 = 1 — this is sin²+cos² = 1, worth knowing
Q5. sin 45° = opposite/hypotenuse → √2/2 = opp/8 → opp = 8 × √2/2 = 4√2 cm
Q6. cos 0° = 1
Q7. tan 60° = √3
📌 Key Facts to Memorise
Memorise the table (angles 0°, 30°, 45°, 60°, 90°):
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan | 0 | 1/√3 (=√3/3) | 1 | √3 | — |
Memory hooks: sin climbs 0, 1, 2, 3 over √-denominators (√0, √1, √2, √3, √4)/2; cos is sin reversed; tan = sin ÷ cos.
🗺️ In the Exam
Exact values are "write down" 1-mark questions on Paper 1 (non-calculator). They also gate trig-graph and exact-area questions later, so the table pays off repeatedly.
Links to: trigonometry ratios (R12/G), trig graphs (A12), Pythagoras (G).
⚡ Targeted Drills
Thirty-second checks — cover the answers, work each one, then mark yourself. If any takes longer than a minute, revisit the matching section above.
- sin 30° → 1/2 · sin 45° → √2/2 · sin 90° → 1
- cos 60° → 1/2 · cos 45° → √2/2 · cos 90° → 0
- tan 30° → 1/√3 · tan 45° → 1 · tan 60° → √3
- sin 30° + cos 60° → 1/2 + 1/2 = 1
- Hypotenuse 4 cm, angle 60°: exact opposite side? → 4 × √3/2 = 2√3 cm
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 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
🧠 Problem-Solving Strategies
Problem-Solving
Memorise exact values: sin 0=0, sin 30=1/2, sin 45=1/root2, sin 60=root3/2, sin 90=1. Cos follows the same values in reverse order. Tan 0=0, tan 30=1/root3, tan 45=1, tan 60=root3. Use the special triangles: 45-45-90 (sides 1:1:root2) and 30-60-90 (sides 1:root3:2).
Multi-Step Problem
Find the exact value of sin(60) x cos(30) + sin(30) x cos(60) without using a calculator.
Solution: sin(60) = root3/2, cos(30) = root3/2, sin(30) = 1/2, cos(60) = 1/2. Result = (root3/2)(root3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1. This equals sin(90) = 1, confirming the addition formula sin(A+B).
⚠️ 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.
✍️ 6-Mark Exam 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.
(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
📊 AO3: Reason & Interpret
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.
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