Master Trigonometric Identities: 60 Flashcards from the Pythagorean Identity to Euler's Formula

Trigonometric identities are usually met as a wall of formulas copied onto a cheat sheet. The Flip It deck JV6351, Trigonometric Identities & Equations (60 cards), is built the other way round: several cards are explicit derivations, so you rebuild an identity before you drill it. Once sin²θ + cos²θ = 1 comes from the unit circle rather than from memory, the rest stops being arbitrary.

Sample cards

The 60 cards, family by family

• Pythagorean and reciprocal core — the definition of an identity, sin²θ + cos²θ = 1 and its two derivations, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ, sin²θ = 1 − cos²θ, cos²θ = 1 − sin²θ, csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ, tan θ = sin θ/cos θ, cot θ = cos θ/sin θ.
• Symmetry and periodicity — sine and tangent odd, cosine even; the three cofunction identities; period 2π for sine and cosine, π for tangent and cotangent.
• Angle sum, difference, multiples — the six sum and difference formulas, sin 2θ, the three forms of cos 2θ, tan 2θ, both half-angle formulas with their ± sign, tan(θ/2) in two equivalent forms, and the triple-angle formulas sin 3θ = 3 sin θ − 4 sin³θ and cos 3θ = 4 cos³θ − 3 cos θ.
• Rewriting products and powers — power reduction for sin²θ, cos²θ, tan²θ; the three product-to-sum formulas; the four sum-to-product formulas.
• Complex exponentials — Euler's formula e^(iθ) = cos θ + i sin θ, Euler's identity e^(iπ) + 1 = 0, cos θ and sin θ written with e^(iθ) and e^(−iθ), De Moivre's formula, and the link to hyperbolic functions: sin(iθ) = i sinh θ, cos(iθ) = cosh θ.
• Applications and curiosities — Ptolemy's theorem and its geometric proof of the sum formulas, prosthaphaeresis, the Weierstrass substitution t = tan(θ/2) for integrals, Morrie's law cos 20° · cos 40° · cos 80° = 1/8, and Machin's formula for π.

Two tips for this specific deck

1. Treat the three Pythagorean identities as one card with two dividers. The deck hands you the derivation: divide sin²θ + cos²θ = 1 by cos²θ and you get tan²θ + 1 = sec²θ; divide by sin²θ and you get 1 + cot²θ = csc²θ. Drill the division, not the two results — one fact instead of three, and it survives exam stress.

2. Split the deck around card 47. Everything before it is the standard curriculum set an exam will actually ask for. Euler, De Moivre, Ptolemy, Morrie, Machin and the half-angle substitution belong to calculus, competition maths and complex analysis: review them in a separate session so their low frequency does not slow down the formulas you need weekly.

Where to find it

Log in at flip-it.app and search for the code JV6351 in the library. If exact values and the unit circle are still shaky, start with Memorize Trigonometry: SOH-CAH-TOA, the Unit Circle and Every Exact Value in 72 Flashcards first — identities are much easier once sin(30°) is automatic.