Quick Summary
Trigonometry introduces the relationship between angles and sides of a right triangle. Class 9 covers the six trigonometric ratios, standard values, complementary angle relationships, and three fundamental identities. Numerical problems and identity proofs are the most common question types.
Trigonometric Ratios
In a right triangle with angle θ, opposite side O, adjacent side A, and hypotenuse H:
| Ratio | Formula | Reciprocal |
|---|---|---|
| sin θ | O/H | cosec θ = H/O |
| cos θ | A/H | sec θ = H/A |
| tan θ | O/A | cot θ = A/O |
Memory aid: "Some Officers Have Curly Black Hair" → sin = O/H, cos = A/H, tan = O/A
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 | undefined |
Complementary Angle Relationships
For complementary angles (A + B = 90°):
- 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
Fundamental Identities
1. sin²θ + cos²θ = 1 → sin²θ = 1 − cos²θ; cos²θ = 1 − sin²θ 2. 1 + tan²θ = sec²θ → tan²θ = sec²θ − 1 3. 1 + cot²θ = cosec²θ → cot²θ = cosec²θ − 1
Important Questions
One Mark
1. If sin θ = 3/5, find cos θ. 2. Find the value of tan 45° × cos 60°. 3. State the identity: sin²θ + cos²θ = ? 4. Simplify: sin(90° − 30°). 5. If tan θ = 1, find θ.
Short Answer — 3 marks
1. In a right triangle, if the opposite side = 5 and hypotenuse = 13, find all six trigonometric ratios. 2. Prove: sin²30° + cos²60° + tan²45° = 3/2. 3. Evaluate: (sin 30° × tan 60°) / (cos 45° × cosec 30°).
Long Answer — 5 marks
1. Prove the identity: (sin θ + cosec θ)² + (cos θ + sec θ)² = 7 + tan²θ + cot²θ. 2. If sin θ = 5/13, find the values of all other trigonometric ratios. Then verify sin²θ + cos²θ = 1.
Common Mistakes
- Using the identity in the wrong form — e.g., writing sin²θ + cos²θ = 1 as sinθ + cosθ = 1.
- Forgetting that tan 90° is undefined (denominator cos 90° = 0).
- Mixing up which side is "opposite" and which is "adjacent" for a given angle.
- Not rationalising surds in the final answer (e.g., leaving 1/√2 instead of √2/2).
Revision Checklist
- [ ] Memorised definitions of all six trigonometric ratios
- [ ] Know the standard values table for 0°, 30°, 45°, 60°, 90°
- [ ] Know all complementary angle identities
- [ ] Memorised the three fundamental identities
- [ ] Can find remaining ratios given one ratio
- [ ] Practised at least three identity proof questions