#businessarticle9 #mathematics #trigonometry

Rules for trigonometry in pure mathematics

By Lona Matshingana 

2026/01/01

# Fundamental Rules of Trigonometry

Trigonometry in pure mathematics is built on a foundation of definitions, identities, and theorems. Here are the core rules and principles:

## Basic Definitions

The trigonometric functions are defined either via the unit circle or as ratios in right triangles:

For an angle θ in a right triangle with opposite side o, adjacent side a, and hypotenuse h:

- sin θ = o/h

- cos θ = a/h  

- tan θ = o/a

- csc θ = h/o

- sec θ = h/a

- cot θ = a/o

## Pythagorean Identities

These follow from the Pythagorean theorem:

- sin²θ + cos²θ = 1

- 1 + tan²θ = sec²θ

- 1 + cot²θ = csc²θ

## Angle Sum and Difference Formulas

- sin(α ± β) = sin α cos β ± cos α sin β

- cos(α ± β) = cos α cos β ∓ sin α sin β

- tan(α ± β) = (tan α ± tan β)/(1 ∓ tan α tan β)

## Double and Half Angle Formulas

Double angles:

- sin 2θ = 2 sin θ cos θ

- cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ

- tan 2θ = 2 tan θ/(1 - tan²θ)

Half angles:

- sin²(θ/2) = (1 - cos θ)/2

- cos²(θ/2) = (1 + cos θ)/2

## Product-to-Sum and Sum-to-Product Formulas

These convert between products and sums of trigonometric functions, useful in integration and Fourier analysis.

Product-to-sum:

- sin α cos β = [sin(α + β) + sin(α - β)]/2

- cos α cos β = [cos(α + β) + cos(α - β)]/2

- sin α sin β = [cos(α - β) - cos(α + β)]/2

## Periodicity and Symmetry

- sin and cos have period 2π; tan has period π

- sin is odd: sin(-θ) = -sin θ

- cos is even: cos(-θ) = cos θ

- tan is odd: tan(-θ) = -tan θ

## Law of Sines and Cosines

For any triangle with sides a, b, c opposite angles A, B, C:

Law of Sines: a/sin A = b/sin B = c/sin C

Law of Cosines: c² = a² + b² - 2ab cos C

These rules form the foundation for solving problems involving angles, triangles, periodic phenomena, and much more in pure mathematics.

Thank you for reading!!!