Study Notes: Introduction to Trigonometry (Class 10)

Overview

Trigonometry forms the bridge between geometry and algebra, dealing with relationships between sides and angles of triangles. For SOF IMO Class 10, this topic carries significant weight as it combines conceptual understanding with computational skill. Questions typically test your grasp of the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent), complementary angle relationships, fundamental identities, and real-world applications through heights and distances problems.

Mastery requires memorizing standard angle values (0°, 30°, 45°, 60°, 90°), fluency with trigonometric identities for simplification, and the ability to set up right triangles from word problems. The Achievers Section often features multi-step problems combining multiple identities or involving complex angle manipulations. Focus on quick recall of ratios and identities, as time management is crucial in the olympiad format.

Key Concepts

• **Trigonometric Ratios** — In a right triangle with angle θ, the six ratios relate the sides: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent, and their reciprocals cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.

• **Domain Restrictions** — Trigonometric ratios are defined only when denominators are non-zero; for example, tan 90° and sec 90° are undefined because cos 90° = 0.

• **Complementary Angles** — Two angles are complementary if they sum to 90°. Key relationships: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ, and vice versa for reciprocals.

• **Pythagorean Identities** — These stem from Pythagoras theorem applied to the unit circle: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ. These identities are essential for simplification and proof problems.

• **Standard Angle Values** — Memorize exact values for 0°, 30°, 45°, 60°, 90° for all six ratios. For instance, sin 30° = 1/2, cos 45° = 1/√2, tan 60° = √3, cosec 90° = 1.

• **Heights and Distances** — Real-world problems involve angle of elevation (looking up) or angle of depression (looking down). The key is to identify the right triangle, label sides relative to the angle, and choose the appropriate trigonometric ratio to solve for the unknown.

• **Ratio Relationships** — Remember that tan θ = sin θ/cos θ and cot θ = cos θ/sin θ. These help convert expressions between different ratios during simplification.

• **Sign Convention** — In Class 10 scope, all angles are acute (0° to 90°) and all trigonometric ratios are positive, simplifying computations and avoiding sign errors.

Formulas / Key Facts

**Basic Ratios:**

• sin θ = perpendicular/hypotenuse, cos θ = base/hypotenuse, tan θ = perpendicular/base
• cosec θ = hypotenuse/perpendicular, sec θ = hypotenuse/base, cot θ = base/perpendicular

**Complementary Angle Formulas:**

• 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:**

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

**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 |

**Reciprocal Relations:**

• cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
• tan θ = sin θ/cos θ, cot θ = cos θ/sin θ

Worked Examples

**Example 1: Simplify (sin²30° + cos²30°) × tan 45°**

Step 1: Substitute standard values. sin 30° = 1/2, cos 30° = √3/2, tan 45° = 1

Step 2: Calculate sin²30° = (1/2)² = 1/4 and cos²30° = (√3/2)² = 3/4

Step 3: Add inside brackets: 1/4 + 3/4 = 1

Step 4: Multiply by tan 45°: 1 × 1 = 1

**Answer: 1**

**Example 2: If sin A = 3/5, find cos A and tan A**

Step 1: Use identity sin²A + cos²A = 1. So (3/5)² + cos²A = 1

Step 2: Calculate 9/25 + cos²A = 1, hence cos²A = 16/25

Step 3: cos A = 4/5 (positive since A is acute)

Step 4: tan A = sin A/cos A = (3/5)/(4/5) = 3/4

**Answer: cos A = 4/5, tan A = 3/4**

**Example 3: A tree casts a 15 m shadow when the angle of elevation of the sun is 60°. Find the height of the tree.**

Step 1: Draw a right triangle where height h is opposite to 60° and shadow 15 m is adjacent.

Step 2: Use tan 60° = h/15, and tan 60° = √3

Step 3: So h/15 = √3, giving h = 15√3

Step 4: Calculate 15 × 1.732 ≈ 25.98 m

**Answer: 15√3 m or approximately 26 m**

Common Mistakes

• **Confusing sin θ and cosec θ** — Students often invert the wrong ratio. Remember: cosec θ is the reciprocal (1/sin θ), not the complementary angle formula. If sin 30° = 1/2, then cosec 30° = 2, not cos 30°.

• **Wrong complementary angle substitution** — Writing sin(90° − 30°) = sin 60° is correct, but some forget to simplify: sin(90° − 30°) = cos 30°. Always apply the complementary formula before plugging in values.

• **Misremembering standard values** — Mixing up sin 60° and cos 60° is common. Use mnemonic: sin increases from 0° to 90° (0 → 1), cos decreases (1 → 0). So sin 60° = √3/2 (larger) and cos 60° = 1/2 (smaller).

• **Forgetting to check undefined values** — Writing tan 90° = something leads to errors. Always note: tan 90°, sec 90°, cosec 0°, cot 0° are undefined because they involve division by zero.

• **Incorrect identity application** — Using sin²θ + cos²θ = 1 correctly but forgetting that 1 + tan²θ = sec²θ (not cosec²θ). Each identity has a specific form; don't mix sec and cosec terms.

Quick Reference

• **Six Ratios:** sin, cos, tan (primary); cosec, sec, cot (reciprocals). Know all definitions from a right triangle.
• **Complementary Rule:** sin(90° − θ) = cos θ and vice versa; extends to all six ratios.
• **Three Pythagorean Identities:** sin² + cos² = 1, 1 + tan² = sec², 1 + cot² = cosec². Memorize cold.
• **Standard Angles:** 0°, 30°, 45°, 60°, 90° — drill the table until instant recall.
• **Heights/Distances Setup:** Identify right triangle, label angle, pick correct ratio (usually tan for angle of elevation/depression).
• **Simplification Strategy:** Convert all ratios to sin and cos using basic definitions and identities, then simplify.