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Complete NCERT Solutions for CBSE Class 10 Maths Chapter 8 — Introduction to Trigonometry: finding trig ratios, evaluating at standard angles, and proving trigonometric identities.
Key Concepts (Quick Recap)
Ratios: sin = opp/hyp, cos = adj/hyp, tan = opp/adj; cosec, sec, cot are their reciprocals.
Standard angle table:
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ |
Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
Exercise 8.1 — Trigonometric Ratios
Q1. If tan A = 4/3, find sin A and cos A. opp = 4, adj = 3 → hyp = √(16 + 9) = 5. So sin A = 4/5, cos A = 3/5.
Q2. If sin A = 3/4, find cos A and tan A. opp = 3, hyp = 4 → adj = √(16 − 9) = √7. So cos A = √7/4, tan A = 3/√7.
Q3. If cot θ = 7/8, evaluate (1 + sin θ)(1 − sin θ) / [(1 + cos θ)(1 − cos θ)]. The expression = (1 − sin²θ)/(1 − cos²θ) = cos²θ/sin²θ = cot²θ = (7/8)² = 49/64.
Exercise 8.2 — Standard Angles
Q1. Evaluate sin 60° cos 30° + sin 30° cos 60°. = (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1.
Q2. Evaluate 2 tan²45° + cos²30° − sin²60°. = 2(1) + (√3/2)² − (√3/2)² = 2 + 3/4 − 3/4 = 2.
Q3. Evaluate (5cos²60° + 4sec²30° − tan²45°) / (sin²30° + cos²30°). Numerator = 5(1/4) + 4(4/3) − 1 = 5/4 + 16/3 − 1 = (15 + 64 − 12)/12 = 67/12. Denominator = 1/4 + 3/4 = 1. Value = 67/12.
Q4. Evaluate sin 60° cos 30° − cos 60° sin 30°. = 3/4 − 1/4 = 1/2 (this equals sin(60° − 30°) = sin 30° = 1/2).
Exercise 8.3 — Trigonometric Identities
Q1. Prove: (1 − cos θ)/(1 + cos θ) = (cosec θ − cot θ)². RHS = (1/sinθ − cosθ/sinθ)² = ((1 − cosθ)/sinθ)² = (1 − cosθ)²/sin²θ = (1 − cosθ)²/(1 − cos²θ) = (1 − cosθ)²/[(1 − cosθ)(1 + cosθ)] = (1 − cosθ)/(1 + cosθ) = LHS. ∎
Q2. Prove: sin θ/(1 + cos θ) + (1 + cos θ)/sin θ = 2 cosec θ. LHS = [sin²θ + (1 + cosθ)²] / [sinθ(1 + cosθ)] = [sin²θ + 1 + 2cosθ + cos²θ] / [sinθ(1 + cosθ)] = [2 + 2cosθ] / [sinθ(1 + cosθ)] = 2(1 + cosθ)/[sinθ(1 + cosθ)] = 2/sinθ = 2 cosec θ = RHS. ∎
Q3. Prove: (1 + tan²A)/(1 + cot²A) = tan²A. LHS = sec²A / cosec²A = (1/cos²A)/(1/sin²A) = sin²A/cos²A = tan²A = RHS. ∎
Common Mistakes
- Mixing up opposite and adjacent sides when reading a ratio.
- Forgetting that the standard-angle table for cos is the reverse of the sin row.
- In identity proofs, jumping between sides — work on one side and transform it into the other.
Exam Tips
- Memorise the 0°–90° table cold; most evaluation questions are direct substitutions.
- For identities, convert everything to sin and cos when stuck.
- Use sin²θ + cos²θ = 1 and its rearrangements (1 − cos²θ = sin²θ) freely.