Class 10 Maths Notes — Chapter 8: Introduction to Trigonometry
Trigonometric ratios of an acute angle, identities and complementary angle relations.
Detailed NCERT notes
- In a right triangle, for an acute angle θ, define: sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent.
- Reciprocals: cosec θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ = cos θ/sin θ.
- Values for standard angles 0°, 30°, 45°, 60°, 90° (memorise the table).
- sin 30° = cos 60° = 1/2; sin 45° = cos 45° = 1/√2; sin 60° = cos 30° = √3/2; tan 45° = 1; tan 30° = 1/√3; tan 60° = √3.
- Fundamental identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
- Trigonometric ratios of complementary angles: sin(90° − θ) = cos θ; tan(90° − θ) = cot θ; sec(90° − θ) = cosec θ.
- Domain caution: tan θ, sec θ undefined at θ = 90°; cot θ, cosec θ undefined at θ = 0°.
- For proving identities, the reliable route: convert all functions to sin and cos, take a common denominator, apply sin² + cos² = 1.
- For 'evaluate' questions, replace using values or complementary-angle substitution before simplifying.
Formulas & key results
- sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ
- sin(90° − θ) = cos θ; tan(90° − θ) = cot θ
- Standard values table for 0°, 30°, 45°, 60°, 90°
Mind map
- Ratios (SOH-CAH-TOA)
- Identities
- Complementary angles
- Standard angle values
Tricks & shortcuts
- Convert to sin/cos when stuck on identity proofs.
- Use complementary angles to collapse cos(90° − θ) into sin θ.
Common mistakes to avoid
- sin²θ ≠ sin(θ²).
- Confusing 1 + tan²θ = sec²θ vs 1 + cot²θ = cosec²θ.
- Forgetting domain (θ ≠ 90° for tan).
Competency-based questions & answers
- Q. Prove (1 + cot A − cosec A)(1 + tan A + sec A) = 2.A. Convert to sin/cos; combine over sin A · cos A; use sin² + cos² = 1; simplifies to 2.
- Q. If sin θ = 3/5, find cos θ and tan θ.A. cos²θ = 1 − 9/25 = 16/25 ⇒ cos θ = 4/5 ⇒ tan θ = 3/4.
- Q. Evaluate: sin 60° cos 30° + sin 30° cos 60°.A. (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1 (= sin 90°).