Trig identities/equations/graphs - Mathematics: Analysis & Approaches IB Study Notes

Fundamental Trigonometric Identities form the backbone of advanced mathematics. The Pythagorean Identity states that sin²θ + cos²θ = 1 for all angles θ, which can be rearranged to give tan²θ + 1 = sec²θ and 1 + cot²θ = csc²θ. These derive from the unit circle definition where coordinates (cosθ, sinθ) satisfy x² + y² = 1.

Compound Angle Formulae allow us to expand trigonometric functions of combined angles:

• sin(A ± B) = sinA cosB ± cosA sinB
• cos(A ± B) = cosA cosB ∓ sinA sinB
• tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB)

Double Angle Formulae are special cases where B = A:

• sin(2θ) = 2sinθ cosθ
• cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
• tan(2θ) = 2tanθ/(1 - tan²θ)

Reciprocal identities define cosecθ = 1/sinθ, secθ = 1/cosθ, and cotθ = 1/tanθ = cosθ/sinθ.

Memory Aid (CAST diagram): In quadrants, Cosine is positive in quadrant 4, All functions positive in quadrant 1, Sine positive in quadrant 2, Tangent positive in quadrant 3.

Trigonometric graphs have distinct characteristics: sine and cosine have amplitude 1, period 2π, with sine passing through the origin while cosine starts at maximum. Tangent has period π with vertical asymptotes at odd multiples of π/2. Transformations follow y = a sin(b(x - c)) + d where a affects amplitude, b affects period (2π/b), c is horizontal shift, and d is vertical shift.

Trigonometric identities and equations model oscillating phenomena throughout nature and technology. Consider sound waves: when two musical notes combine, the resulting waveform uses compound angle formulae. A piano playing middle C (262 Hz) and E (330 Hz) simultaneously creates a complex wave described by sin(262·2πt) + sin(330·2πt), which musicians simplify using sum-to-product identities to understand beats and harmonics.

Engineering applications utilize trigonometric equations extensively. Electrical engineers model alternating current as I(t) = I₀ sin(ωt + φ), where solving for specific current values requires trigonometric equations. When designing bridges, structural engineers use double angle formulae to calculate stress distributions on angled supports.

Satellite communication relies on solving trigonometric equations to determine optimal transmission angles. GPS technology triangulates position by solving systems involving multiple trigonometric relationships simultaneously.

Think of tan(2θ) = 2tanθ/(1 - tan²θ) as "doubling the angle doesn't double the tangent" — analogous to how doubling your speed doesn't double your journey time if distance changes. The Pythagorean identity resembles a financial budget where sin²θ and cos²θ are "allocations" that must sum to 1 (100%) regardless of the angle.

Oceanographers model tide patterns using combinations of sine waves with different periods, requiring mastery of compound angles to predict high tides accurately. Astronomers track planetary positions using trigonometric functions, solving equations to predict eclipses and conjunctions centuries in advance. Understanding these real applications transforms abstract identities into powerful tools for modeling cyclical phenomena across disciplines.