Introduction to Trigonometry — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Introduction to Trigonometry (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Introduction to Trigonometry (CBSE & HBSE)
One of the most important chapters — typically 8-12 marks. CBSE tests trigonometric ratios, identities, and complementary angle relations. HBSE includes direct ratio computation and identity proofs.
Trigonometric Ratios
Trigonometric Ratios of an Acute Angle
In a right-angled triangle with acute angle θ:
| Ratio | Full Name | Formula | Abbreviation |
|---|---|---|---|
| sin θ | Sine | Opposite/Hypotenuse | P/H |
| cos θ | Cosine | Adjacent/Hypotenuse | B/H |
| tan θ | Tangent | Opposite/Adjacent | P/B |
| cosec θ | Cosecant | Hypotenuse/Opposite | H/P |
| sec θ | Secant | Hypotenuse/Adjacent | H/B |
| cot θ | Cotangent | Adjacent/Opposite | B/P |
Memory Aid: "Pandit Badri Prasad Har Har Bole" → sin=P/H, cos=B/H, tan=P/B
Reciprocal Relationships
- sin θ × cosec θ = 1 → cosec θ = 1/sin θ
- cos θ × sec θ = 1 → sec θ = 1/cos θ
- tan θ × cot θ = 1 → cot θ = 1/tan θ
- tan θ = sin θ/cos θ, cot θ = cos θ/sin θ
Standard Values Table
| Angle | 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 | Not defined |
| cosec | Not def | 2 | √2 | 2/√3 | 1 |
| sec | 1 | 2/√3 | √2 | 2 | Not defined |
| cot | Not def | √3 | 1 | 1/√3 | 0 |
Memory Trick for sin: 0, 1, 2, 3, 4 → sin = 0/2, 1/2, 2/2, 3/2, 4/2 → 0, 1/2, 1/√2, √3/2, 1
IMPORTANT: tan 90° and cot 0° are NOT defined (division by zero).
Trigonometric Identities
Three Fundamental Identities
$$\sin^2\theta + \cos^2\theta = 1$$ $$1 + \tan^2\theta = \sec^2\theta$$ $$1 + \cot^2\theta = \text{cosec}^2\theta$$
Derived from first identity:
- sin²θ = 1 - cos²θ
- cos²θ = 1 - sin²θ
- sec²θ - tan²θ = 1
- cosec²θ - cot²θ = 1
Proving Trigonometric Identities
Strategy:
- Take the more complex side (LHS)
- Convert everything to sin and cos
- Simplify using algebra and identities
- Arrive at the other side (RHS)
Example: Prove: (1 + tan²A)/(1 + cot²A) = tan²A
- LHS = sec²A / cosec²A = (1/cos²A)/(1/sin²A) = sin²A/cos²A = tan²A = RHS ✓
Common Identity Proof Techniques
| Technique | When to Use |
|---|---|
| Convert to sin/cos | Always useful to start |
| Multiply by conjugate | Denominators like (1+sinA), (secA-tanA) |
| Factor a³-b³ or a³+b³ | Powers of 3 in question |
| sin²θ+cos²θ=1 substitution | When both sin and cos present |
Conjugate pairs to remember:
- (1+sinA)(1-sinA) = cos²A
- (secA+tanA)(secA-tanA) = 1
- (cosecA+cotA)(cosecA-cotA) = 1
CBSE Pattern: Identity proofs are 3-4 marks. Start with LHS, never cross-multiply, end at RHS.
Complementary Angles in Trigonometry
Complementary Angles
Angles A and B are complementary if A + B = 90°.
If A + B = 90°, then B = 90° - A.
Trigonometric Ratios of Complementary Angles
| Function | Complementary Relation |
|---|---|
| sin(90°-θ) | = cos θ |
| cos(90°-θ) | = sin θ |
| tan(90°-θ) | = cot θ |
| cot(90°-θ) | = tan θ |
| sec(90°-θ) | = cosec θ |
| cosec(90°-θ) | = sec θ |
Memory: Functions swap between 'co' pairs: sin↔cos, tan↔cot, sec↔cosec.
Applying Complementary Angles
Type 1: Evaluate expressions like sin 27°/cos 63°
- 63° = 90° - 27° → cos 63° = sin 27°
- sin 27°/sin 27° = 1
Type 2: Find θ in sin(θ+6°) = cos(3θ-24°)
- cos(3θ-24°) = sin(90°-(3θ-24°)) = sin(114°-3θ)
- So θ+6° = 114°-3θ → 4θ = 108° → θ = 27°
Type 3: Simplify using pairs (angles summing to 90°) Example: sin 34°/cos 56° + cos 34°/sin 56°
- 56° = 90°-34° → cos 56° = sin 34°, sin 56° = cos 34°
- = sin 34°/sin 34° + cos 34°/cos 34° = 1 + 1 = 2
CBSE Tip: Always look for pairs of angles summing to 90°. This converts unfamiliar angles to standard ones.
Frequently asked questions
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Do these notes follow CBSE and HBSE?
Yes. The Introduction to Trigonometry notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
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Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Introduction to Trigonometry.