Maths
Trigonometry — sin, cos, tan & identities — quick study summary
GCSE MathsA-Level MathsAP PrecalculusIB Maths
Trigonometry studies the relationships between angles and sides of triangles. In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOH CAH TOA). The unit circle extends these to any angle. Pythagorean identity: sin²θ + cos²θ = 1. Law of sines (a/sinA = b/sinB = c/sinC) and law of cosines (c² = a² + b² − 2ab·cosC) solve any triangle.
Key points
- SOH CAH TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj
- Unit circle: sin θ = y-coordinate, cos θ = x-coordinate of point at angle θ
- Identity: sin²θ + cos²θ = 1
- Law of sines: a/sinA = b/sinB = c/sinC
- Law of cosines: c² = a² + b² − 2ab·cosC (generalises Pythagoras)
Practice quiz
Click each question to reveal the answer.
1. In a right triangle, the opposite side to angle θ is 3 and the hypotenuse is 5. What is sin θ?
- 3/5
- 4/5
- 5/3
- 3/4
Answer: 3/5
sin = opposite/hypotenuse = 3/5.
2. If sin θ = 0.6, what is cos θ (assuming θ is acute)?
Answer: 0.8
sin² + cos² = 1 → cos² = 1 − 0.36 = 0.64 → cos = 0.8 (positive since θ is acute).
3. What is tan(45°)?
Answer: 1
At 45°, opposite = adjacent in a right triangle, so tan = 1.
Last reviewed: May 2026