Introduction to Trigonometry

Why This Chapter Matters

Trigonometry is tested EVERY year — typically 8-12 marks in boards. Trig ratios, identities, and complementary angles all appear regularly. The identities are especially important as they require proof and simplification skills. Chapter 9 (Applications) is a continuation.

Prerequisites

Core Concepts

1. Trigonometric Ratios

For a right-angled triangle with angle θ (theta):

           Hypotenuse (H)
          /|
         / |
        /  | Opposite (O) — side opposite to θ
       /θ  |
      /____| 
  Adjacent (A) — side adjacent to θ

|---|---|---|

Important: tan θ = sin θ / cos θ | cot θ = cos θ / sin θ

2. Trigonometric Ratios of Standard Angles

|---|---|---|---|---|---|

Memory for sin: 0, 1/2, 1/√2, √3/2, 1 → divide √(0,1,2,3,4) by 2

cos is reverse of sin: 1, √3/2, 1/√2, 1/2, 0

3. Trigonometric Identities (Most Important for Board!)

Identity 1: sin²θ + cos²θ = 1

Derived forms:

Identity 2: 1 + tan²θ = sec²θ

Derived forms:

Identity 3: 1 + cot²θ = cosec²θ

Derived forms:

How to prove an identity: Take one side (usually LHS), simplify step by step until you reach RHS. Never operate on both sides simultaneously.

4. Complementary Angles

Two angles are complementary if their sum = 90°.

If θ is an angle, its complement is (90° − θ).

Key relationships:

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

Trick: sin↔cos, tan↔cot, sec↔cosec swap with complementary angles.

Solved Examples

Example 1 — Finding All Ratios

Q: If sin A = 3/4, find all other trigonometric ratios.

sin A = O/H = 3/4 → Opposite = 3, Hypotenuse = 4

By Pythagoras: Adjacent = √(16−9) = √7

cos A = √7/4 | tan A = 3/√7 | cosec A = 4/3 | sec A = 4/√7 | cot A = √7/3

Example 2 — Proving Identity

Q: Prove: (sinθ + cosecθ)² + (cosθ + secθ)² = 7 + tan²θ + cot²θ

LHS = sin²θ + 2sinθcosecθ + cosec²θ + cos²θ + 2cosθsecθ + sec²θ

= (sin²θ + cos²θ) + 2(1) + cosec²θ + 2(1) + sec²θ

= 1 + 4 + (1 + cot²θ) + (1 + tan²θ)

= 7 + tan²θ + cot²θ = RHS □

Example 3 — Complementary Angles

Q: Evaluate: tan 65° / cot 25°

cot 25° = cot(90° − 65°) = tan 65°

→ tan 65° / tan 65° = 1

PYQs

2023

Q: If cosecθ = 13/12, find sinθ + cosθ.

sinθ = 12/13, cosθ = √(1 − 144/169) = 5/13

sinθ + cosθ = 12/13 + 5/13 = 17/13

2022

Q: Prove: (1 + cotA − cosecA)(1 + tanA + secA) = 2

LHS: multiply out, use identities, simplify to 2

2021

Q: If tanθ + 1/tanθ = 2, find tan²θ + 1/tan²θ

(tanθ + 1/tanθ)² = 4 → tan²θ + 2 + 1/tan²θ = 4 → tan²θ + 1/tan²θ = 2

2020

Q: Evaluate: sin²25° + sin²65° + √3 tan5°·tan85°

= sin²25° + cos²25° + √3·tan5°·cot5° = 1 + √3·1 = 1 + √3

MCQ Practice

Q1. If sinA = 1/2, then 3cosA − 4cos³A =

(A) 1 (B) 0 ✓ (C) 1/2 (D) √3/2

[A = 30°, 3cos30° − 4cos³30° = 3(√3/2) − 4(3√3/8) = 3√3/2 − 3√3/2 = 0]

Q2. sec²10° − cot²80° =

(A) 0 (B) 1 ✓ (C) −1 (D) 2

[cot80° = cot(90°−10°) = tan10°, so sec²10° − tan²10° = 1]

Q3 (Hard). If cosθ + cos²θ = 1, then sin¹²θ + 3sin¹⁰θ + 3sin⁸θ + sin⁶θ + 2sin⁴θ + 2sin²θ − 2 = ?

[cosθ = 1 − cos²θ = sin²θ, so cos²θ = sin⁴θ, then simplify using substitution: answer = 1]

Revision Notes

SOHCAHTOA:
  sin = Opp/Hyp    cos = Adj/Hyp    tan = Opp/Adj

Three Fundamental Identities:
  sin²θ + cos²θ = 1
  1 + tan²θ = sec²θ
  1 + cot²θ = cosec²θ

Complementary Angle pairs (add to 90°):
  sin ↔ cos  |  tan ↔ cot  |  sec ↔ cosec

Standard angles: 0, 30, 45, 60, 90 degrees
sin: 0, 1/2, 1/√2, √3/2, 1
cos: 1, √3/2, 1/√2, 1/2, 0

Identity Proof Strategy:

Common Mistakes:

❌ sin²θ + cos²θ = 1, NOT 2

❌ sin(A+B) ≠ sinA + sinB

❌ Forgetting absolute value for: √(sin²θ) = |sinθ|, not just sinθ

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