Description

TRIGONOMETRY – MATHEMATICS F2
Topic 08: Trigonometry
Form 2 Mathematics Notes | Comprehensive & Exam-Focused

📘 OVERVIEW
This comprehensive set of notes covers Trigonometry – a core topic in Form 2 Mathematics. It is carefully structured to help students understand trigonometric ratios, special angles, trigonometric ratios of any angle, relationships between ratios, angles of elevation and depression, and their real-life applications, with clear worked examples and practice questions for exam preparation.

✨ WHAT THIS DOCUMENT COVERS
1. Introduction to Trigonometry
Understanding trigonometry

Real-life applications: building constructions, designing structures, navigation and GPS, astronomy, engineering, surveying, photography

2. Trigonometric Ratios
The three basic ratios: sine (sin), cosine (cos), tangent (tan)

Right-angled triangle reference

SOH CAH TOA mnemonic:

SOH: Sin = Opposite/Hypotenuse

CAH: Cos = Adjacent/Hypotenuse

TOA: Tan = Opposite/Adjacent

Reciprocals of basic ratios:

Secant (sec) = 1/cos

Cosecant (csc) = 1/sin

Cotangent (cot) = 1/tan

Worked Examples:

Example 8.1: Label sides and write trigonometric ratios

Example 8.2: Given sin X = YZ/XZ, determine other ratios

Example 8.3: Find sin A, cos A, tan A

Example 8.4: Find QR, tan R, cos R

Example 8.5: Find cos x + sin y

Exercise 8.1 Practice (Q1–Q8)

3. Trigonometric Ratios of Special Angles
Special angles: 30°, 45°, 60°, 90°, 0°, 180°, 270°, 360°

Deriving 30° and 60° Values: Equilateral triangle with side length 2 units

Deriving 45° Values: Isosceles right triangle with legs of 1 unit

Values for 0° and 90°

Summary Table of Special Angles

Worked Examples:

Example 8.6: Find sin 45° + cos 60°

Example 8.7: Find RQ in triangle PQR

Example 8.8: Find 2sin 60° + cos 30°

Example 8.9: Find RQ in isosceles triangle

Exercise 8.2 Practice (Q1–Q8)

4. Trigonometric Ratios of Any Angle
The Four Quadrants

Signs in Each Quadrant (ASTC Rule):

Quadrant I (0°–90°): All positive

Quadrant II (90°–180°): Sin positive

Quadrant III (180°–270°): Tan positive

Quadrant IV (270°–360°): Cos positive

Worked Examples:

Example 8.11: Write signs of trigonometric ratios

Example 8.12: Express in terms of acute angle

Example 8.13: Find positive or negative angle

Example 8.14: Find sine, cosine, and tangent

Exercise 8.3 Practice (Q1–Q3)

5. Relationship Between Trigonometric Ratios
Complementary Angles:

sin(90° − θ) = cos θ

cos(90° − θ) = sin θ

tan(90° − θ) = cot θ = 1/tan θ

Key Identities:

sin²θ + cos²θ = 1
tan θ = sin θ/cos θ
cot θ = cos θ/sin θ = 1/tan θ
sec θ = 1/cos θ
csc θ = 1/sin θ
sin²θ = 1 − cos²θ
cos²θ = 1 − sin²θ
Worked Examples:

Example 8.15: Given sin θ = 4/9, find cos θ and tan θ

Example 8.16: α and β are complementary, sin α = 5/13. Find tan β

Example 8.17: Given sin θ = 0.9397 and cos θ = 0.3420, find tan θ

Exercise 8.4 Practice (Q1–Q10)

6. Calculating Trigonometric Ratios Using a Calculator
Using calculator: sin, cos, tan buttons

Steps: Ensure calculator is in degree mode (DEG), enter angle, press function button

Activity 8.3

7. Inverse of Trigonometric Ratios
Definition: Inverse trigonometric functions give the angle when the ratio is known

Notation: sin⁻¹(x), cos⁻¹(x), tan⁻¹(x)

Important notes:

sin⁻¹ and cos⁻¹ results are between −90° and 90°

sin θ and cos θ cannot exceed 1 in absolute value

tan θ can be any real number

Worked Examples:

Example 8.18: If tan x = 1.4071, find x

Example 8.19: Find x and y correct to 3 significant figures

Example 8.20: Ladder problem

Exercise 8.5 Practice (Q1–Q5)

8. Angles of Elevation and Depression
Definitions:

Angle of Elevation: Angle formed between horizontal line and line of sight when looking upwards

Angle of Depression: Angle formed between horizontal line and line of sight when looking downwards

Key relationship: Angle of Elevation = Angle of Depression (when two lines are parallel)

Worked Examples:

Example 8.21: Find height of tower

Example 8.22: Find PQ

Example 8.23: Find height of mast

Example 8.24: Find angle of elevation of sun

Example 8.25: Find distance west and north

Example 8.26: Find AC and height

Exercise 8.6 Practice (Q1–Q10)

9. Chapter Summary
Basic trigonometric ratios

Reciprocal ratios

Special angles table

Key identities

Quadrant signs (ASTC)

Angles of elevation and depression

10. Revision Exercise 8: Selected Solutions
Q1–Q22 with detailed solutions

🎯 WHY THIS RESOURCE IS PERFECT FOR YOU
✅ Well-Organised – Content is broken down into clear, digestible sections
✅ Exam-Focused – Includes worked examples, practice questions, and revision exercises
✅ Comprehensive – Covers trigonometric ratios, special angles, any angle, relationships, inverse ratios, and real-life applications
✅ Student-Friendly – Simple language with detailed explanations and step-by-step solutions
✅ Revision-Ready – Summary points, formula references, and practice questions
✅ Teacher-Approved – Ideal for classroom use, homework, and exam preparation

📚 WHO IS THIS FOR?
Form 2 Students studying Mathematics

Teachers preparing lesson notes and assessments

Parents supporting their children’s learning

Anyone interested in trigonometry and its applications

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This is a must-have resource for anyone serious about mastering Form 2 Mathematics. Whether you’re revising for exams or teaching in class, these notes will save you time and help you achieve excellent results.

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