Geometric Bites

The associative, commutative and distributive laws are three of the most important structural rules in algebra. They explain how expressions may be grouped, reordered, expanded and simplified without changing their mathematical value. These laws are used throughout arithmetic, algebra, factorisation, equation solving and mathematical proof. Associative Law The associative law describes how terms may be grouped when the same operation is repeated. If an operation is associative, changing the placement of the brackets does not change the final value of the expression. For addition: a + (b + c) = (a + b) + c For example: 1 + (2 + 3) = (1 + 2) + 3 The associative law also applies to multiplication: a × (b × c) = (a × b) × c For example: 2 × (3 × 4) = (2 × 3) × 4 Subtraction is not associative because changing the grouping can change the result. a − (b − c) ≠ (a − b) − c For example: 1 − (2 − 3) ≠ (1 − 2) − 3 Commutative Law The commutative law describe...

These workings use compound angle identities to derive double angle, triple angle, and related trigonometric identities. Compound Angles, Extras 1. Deriving sin(2θ) sin(θ + θ) = sinθ cosθ + cosθ sinθ = 2sinθ cosθ = sin(2θ) 2. Deriving cos(2θ) cos(θ + θ) = cosθ cosθ - sinθ sinθ = cos 2 θ - sin 2 θ = cos(2θ) 3. Deriving cos(2θ) = 2cos²θ - 1 cos(2θ) = cos 2 θ - sin 2 θ = cos 2 θ - (1 - cos 2 θ) = cos 2 θ - 1 + cos 2 θ = 2cos 2 θ - 1 4. Deriving cos(2θ) = 1 - 2sin²θ cos(2θ) = cos 2 θ - sin 2 θ = 1 - sin 2 θ - sin 2 θ = 1 - 2sin 2 θ 5. Deriving sin(3θ) sin(2θ + θ) = sin(2θ)cosθ + cos(2θ)sinθ = 2sinθ cosθ cosθ + (1 - 2sin 2 θ)sinθ = 2sinθ cos 2 θ + sinθ - 2sin 3 θ = sinθ(2cos 2 θ + 1) - 2sin 3 θ = sinθ(2(1 - sin 2 θ) + 1) - 2sin 3 θ = sinθ(2 - 2sin 2 θ + 1) - 2sin 3 θ = sinθ(3 - 2sin 2 θ) - 2sin 3 θ = 3sinθ - 2sin 3 θ - 2sin 3 θ = 3sinθ - 4sin 3 θ 6. Deriving cos(3θ) cos(2θ + θ) = cos2θ cosθ - sin2...

These workings derive the laws of base-10 logarithms from exponent laws by converting between exponential form and logarithmic form. Assume a > 0 , b > 0 , and n ≠ 0 . Product Rule log(ab) = log a + log b Let 10 x = a Let 10 y = b Therefore: log 10 a = x log 10 b = y 10 x 10 y = ab 10 x+y = ab Therefore: log 10 (ab) = x + y Substituting: log 10 (ab) = log 10 a + log 10 b Therefore: log(ab) = log a + log b Quotient Rule log(a / b) = log a - log b Let 10 x = a Let 10 y = b Therefore: log 10 a = x log 10 b = y 10 x / 10 y = a / b 10 x-y = a / b Therefore: log 10 (a / b) = x - y Substituting: log 10 (a / b) = log 10 a - log 10 b Therefore: log(a / b) = log a - log b Power Rule log(a n ) = n log a Let log(a n ) = x Therefore: 10 x = a n Taking the n-th root of both sides: (10 x ) 1/n = (a n ) 1/n 10 x/n = a Therefore: log 10 a = x / n n log 10 a = x Theref...