Trigonometry
Introduction
Trigonometry is the study of relationships between sides and angles of triangles. The word comes from Greek: 'trigonon' (triangle) and 'metron' (measure). For ICSE Class 9, we focus on basic trigonometric ratios and their applications.
Trigonometric Ratios
In a right triangle ABC with right angle at B:
Let angle at A = θ. Then:
- Side opposite to θ = BC = perpendicular (p)
- Side adjacent to θ = AB = base (b)
- Hypotenuse = AC = h
Six Trigonometric Ratios:
| Ratio | Name | Formula |
|---|---|---|
| sin θ | Sine | opposite/hypotenuse = p/h |
| cos θ | Cosine | adjacent/hypotenuse = b/h |
| tan θ | Tangent | opposite/adjacent = p/b |
| cot θ | Cotangent | adjacent/opposite = b/p |
| sec θ | Secant | hypotenuse/adjacent = h/b |
| cosec θ | Cosecant | hypotenuse/opposite = h/p |
Relationships:
- tan θ = sin θ / cos θ
- cot θ = 1 / tan θ
- cot θ = cos θ / sin θ
- sec θ = 1 / cos θ
- cosec θ = 1 / sin θ
sin A = BC/AC = 4/5 cos A = AB/AC = 3/5 tan A = BC/AB = 4/3 </Solution> </ICSEExample>
Standard Angles
The trigonometric ratios for 0°, 30°, 45°, 60°, and 90° must be memorised.
| Angle θ | sin θ | cos θ | tan θ | cot θ | sec θ | cosec θ |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | Not defined | 1 | Not defined |
| 30° | 1/2 | √3/2 | 1/√3 | √3 | 2/√3 | 2 |
| 45° | 1/√2 | 1/√2 | 1 | 1 | √2 | √2 |
| 60° | √3/2 | 1/2 | √3 | 1/√3 | 2 | 2/√3 |
| 90° | 1 | 0 | Not defined | 0 | Not defined | 1 |
Simple Trigonometric Identities
Identity 1: sin²A + cos²A = 1
Proof: In right triangle ABC, right-angled at B: sin A = BC/AC, cos A = AB/AC sin²A + cos²A = BC²/AC² + AB²/AC² = (BC² + AB²)/AC² = AC²/AC² = 1
Identity 2: 1 + tan²A = sec²A
Identity 3: 1 + cot²A = cosec²A
<ICSEExample title="Using Identity"> If sin A = 3/5, find cos A using identity. <Solution> sin²A + cos²A = 1 (3/5)² + cos²A = 1 9/25 + cos²A = 1 cos²A = 16/25 cos A = 4/5 (positive since A is acute) </Solution> </ICSEExample>Simple 2D Problems
<ICSEExample title="Height of a Pole"> A pole casts a shadow of length 10√3 m when the suns elevation is 30°. Find the height of the pole. <Solution> Let height of pole = h tan 30° = h/(10√3) 1/√3 = h/(10√3) h = (10√3)/√3 = 10 m Height of pole = 10 m </Solution> </ICSEExample> <ICSEExample title="Distance Problem"> A ladder leaning against a wall makes an angle of 60° with the ground. If the foot of the ladder is 2.5 m from the wall, find the length of the ladder. <Solution> cos 60° = adjacent/hypotenuse = 2.5/length 1/2 = 2.5/length Length = 5 m </Solution> </ICSEExample>Common Mistakes With Fixes
| Mistake | Correction |
|---|---|
| Confusing sin and cos ratios | Sin = opposite/hypotenuse, Cos = adjacent/hypotenuse |
| Forgetting that sin²A means (sin A)² | sin²A = (sin A)², not sin(A²) |
| Mixing up standard angle values | Draw a quick table or derive using special triangles |
| Using degrees in calculators incorrectly | Ensure calculator is in degree mode |
ICSE Exam Focus
| Topic | Marks (approx.) | Frequency |
|---|---|---|
| Trigonometric ratios of standard angles | 3-4 marks | Very common |
| Trigonometric identities | 4 marks | Very common |
| Simple 2D problems | 4-5 marks | Common |
| Evaluating expressions with trig ratios | 3 marks | Frequently asked |
Self-Test
Q1: If cos A = 12/13, find sin A and tan A.
Q2: Evaluate: cos² 45° + sin² 45°
Q3: Prove: (1 - sin²A) × sec²A = 1
Q4: A tree breaks due to storm and the broken part bends so that the top touches the ground making an angle of 60° with the ground. The distance from the foot of the tree to the point where the top touches the ground is 6 m. Find the height of the tree.
Q5: Find the value of: (sin 30° × cos 60°) + (cos 30° × sin 60°)
