Description

TRIGONOMETRY — Complete Study Guide (006)
Advanced Level Mathematics — Form Five

A comprehensive, exam-ready resource covering everything from right-triangle ratios to inverse trigonometric functions and their graphs — all in one clearly organized document.

📘 What’s Inside
1. Introduction to Trigonometry

Definition and real-life applications (surveying, navigation, astronomy, engineering, physics, computer graphics)

Key concept: the right-angled triangle with all three primary ratios

2. Trigonometric Ratios

All six functions defined: sine, cosine, tangent, cosecant, secant, cotangent — with reciprocal relationships

3. Ratios for Special Angles

30°–60°–90° and 45°–45°–90° triangles fully labeled

Complete summary table (0°, 30°, 45°, 60°, 90°)

4. Signs of Trigonometric Functions

ASTC Rule (All Students Take Coffee) with quadrant diagram

Three worked examples determining signs in different quadrants

5. Trigonometric Identities

Pythagorean identities (all three)

Full proof of sin²θ + cos²θ = 1

Four worked examples: finding ratios from a given ratio, proving the identity (sinθ + cosθ)² = 1 + 2sinθcosθ

6. Compound Angle Formulae

All six formulae: sin(A±B), cos(A±B), tan(A±B)

Visual derivation of sin(A+B)

Three exact-value examples: sin 75°, cos 15°, tan 75°

7. Double Angle Formulae

All forms including cos 2A in three variations

Worked examples: sin 2θ, cos 2θ, and proving identities

8. Half Angle Formulae

All formulae including the useful tan(A/2) rational forms

Worked examples: sin 15° and cos 22.5° using half-angle formulae

9. Trigonometric Equations

Principal solutions and general solution formula table (sin, cos, tan)

Worked examples: solving within a range and finding general solutions

10. Equations of the Form a cos θ + b sin θ = c

The R-formula method (R = √(a² + b²), tan α = b/a)

Fully worked example: 3cosθ + 4sinθ = 5

The t-formula method with a quadratic in tan(θ/2)

11. Factor Formulae

Sum-to-product formulae (all four)

Product-to-sum formulae (all four)

Worked examples including a full proof of the sum-to-product identity

12. Radians and Small Angles

Degree–radian conversion formulae

Complete conversion table (0° to 360°)

Small angle approximations: sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ

Worked example: approximating sin 5°

13. Trigonometric Functions and Graphs

Domain and range table for all six functions

Complete graphs of sin x, cos x, and tan x with key features (amplitude, period, zeros, asymptotes)

Worked examples: finding amplitude/period and graphing a transformed sine curve

14. Inverse Trigonometric Functions

Definitions and important relationships

Worked examples evaluating sin⁻¹, cos⁻¹, and tan⁻¹

Complete graphs of sin⁻¹x, cos⁻¹x, and tan⁻¹x with domains and ranges

Worked examples: sin(cos⁻¹x), cos(tan⁻¹x), simplifying expressions, and proving inverse identities

15. Chapter Summary

Key formulas summary table (Pythagorean, compound, double angle, inverse functions)

Special angle values reference

Inverse function domains and ranges

16. Practice Questions

Seven sections: Trig Ratios & Identities, Compound Angles, Double & Half Angles, Trigonometric Equations, Sum-to-Product, Graphs, and Inverse Functions

✅ Why This Guide Works
Every identity proven — Pythagoras, compound angles, and sum-to-product derived step by step

All graph shapes included — sin, cos, tan and their inverses, with key features labeled

Multiple solving methods — e.g., both the R-formula and t-formula for a cosθ + b sinθ = c

Exam-focused — general solution formulas, conversion tables, and practice questions by topic

Complete coverage — from basic ratios to inverse trigonometric graphs in a single document

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