Key Terms
- Exact value: An answer expressed as a fraction or surd — no rounding
- Surd: An irrational square root left in exact form (e.g. √2, √3)
- 45-45-90 triangle: The special isosceles right triangle giving values for 45°
- 30-60-90 triangle: Half an equilateral triangle giving values for 30° and 60°
Must-Know Facts (memorise these)
- sin 0° = 0, sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1
- cos is sin in REVERSE: cos 0° = 1, cos 30° = √3/2, cos 60° = ½, cos 90° = 0
- tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° = undefined
- Identity: sin²θ + cos²θ = 1 (for any angle)
Key Methods
- Sin pattern: √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90°
- Cos = sin reversed
- Tan = sin ÷ cos
- Simplify √2/2 × √2/2 = 2/4 = ½
Key Formulas
- sin 0° = 0, sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1
- cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½, cos 90° = 0
- tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° = undefined
Common Mistakes
- Confusing sin and cos values: cos values are sin values in reverse order — use the pattern to check
- tan 90° has no value: tan 90° is undefined — do not write 0 or infinity
- Leaving surd form: When an exact answer is required, write √3/2 not 0.866...
- Simplifying incorrectly: 1/√3 should be rationalised to √3/3 for exact form
This topic summary covers Knowledge Organiser: Exact Trig Values within Exact Trig Values for GCSE Mathematics. Revise Exact Trig Values in Geometry & Measures for GCSE Mathematics with 14 exam-style questions and 4 flashcards. This topic shows up very often in GCSE exams, so students should be able to explain it clearly, not just recognise the term. It is section 7 of 7 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Exact Trig Values
What is the exact value of sin 30°?
James says that sin 30° and cos 60° are equal. Is James correct? Explain your answer using exact values.
Quick recall flashcards
Sin Pattern
sin 0°=0, sin 30°=1/2, sin 45°=√2/2, sin 60°=√3/2, sin 90°=1. Pattern: √n/2 for n=0,1,2,3,4
Cos Pattern
Cos is sin BACKWARDS. cos θ = sin(90°-θ). So cos 30° = sin 60°, etc.