Memorize Trigonometry: SOH-CAH-TOA, the Unit Circle and Every Exact Value in 72 Flashcards

Trigonometry is a memory subject disguised as a math subject. Once you can recall sin(30°) = 1/2 and cos(45°) = √2/2 without thinking, identities, the unit circle and the Law of Cosines stop being scary and become bookkeeping. The Trigonometry: SOH-CAH-TOA & Unit Circle deck (code SP9746, 72 cards) on Flip It is built for that recall. This guide shows how to memorize it in an order that sticks.

Sample cards from the deck

Everything the 72 cards cover

• Six ratios: sine, cosine, tangent, cosecant, secant, cotangent, plus tan = sin/cos and cot = cos/sin.
• Exact values: sin, cos and tan of 0°, 30°, 45°, 60°, 90°, and the radian equivalents π/6, π/4, π/3.
• Three Pythagorean identities: sin² + cos² = 1, 1 + tan² = sec², 1 + cot² = csc².
• Function behaviour: periods (2π for sine and cosine, π for tangent), ranges, even/odd symmetry (cos is even, sin and tan are odd), complementary-angle relations and why "co-" functions are named that way.
• Inverse functions: domains and ranges of arcsine, arccosine, arctangent.
• Triangle laws: Law of Sines, Law of Cosines, Law of Tangents, area from two sides and an included angle.
• Beyond the basics: Euler's formula e^(iθ) = cos θ + i·sin θ, the Maclaurin series for sin(x), versine, haversine, chord, coversine, exsecant.
• History and uses: the Greek roots trigōnon + metron, Nasir al-Din al-Tusi, Sumerian 360° circles, Gemma Frisius and triangulation, the sextant, Fourier transforms, GPS, CT scans, ultrasound.

The order that works

First: the 15 exact values. Don't memorize them as 15 facts. Use the hand trick the deck's values follow: for 0°, 30°, 45°, 60°, 90°, sine is √0/2, √1/2, √2/2, √3/2, √4/2 — i.e. 0, 1/2, √2/2, √3/2, 1. Cosine is the same list reversed. Tangent is sine over cosine, which gives you 0, √3/3, 1, √3 and "undefined" for free. Flip the value cards until you can produce all 15 in under a minute.

Second: the unit circle as a picture, not a list. Card 8 is the key: a point on the circle is (cos θ, sin θ). Every later fact — sin² + cos² = 1 is just the circle equation x² + y² = 1; cos(−θ) = cos(θ) because reflecting across the x-axis keeps x; sine has period 2π because one lap is 2π — becomes a consequence of that one card.

Third: identities and laws. Learn 1 + tan² = sec² by dividing the fundamental identity by cos²; learn 1 + cot² = csc² by dividing by sin². The Law of Cosines reduces to Pythagoras when C = 90° (cos 90° = 0), a quick sanity check you can run in an exam.

Last: inverse ranges and the historical cards. Arcsine and arctangent both live in (−π/2, π/2) territory; arccosine alone uses [0, π]. The history cards (al-Tusi, Frisius, Euler, Sumerians) are pure recall — ten minutes a day is plenty.

Two tips specific to this deck

• Say "co-" out loud. Card 52 explains it: cosine, cotangent and cosecant are the sine, tangent and secant of the complementary angle. That single idea covers sin(θ) = cos(90° − θ) and prevents mixing up which reciprocal goes with which function.
• Check parity before you answer any negative-angle card. Even → cos(−θ) = cos θ. Odd → sin and tan flip sign. Three cards, one rule.

Deck SP9746 is waiting in your Flip It library. For the triangle facts that trigonometry builds on — centroids, congruence, polygon angles — see our geometry flashcard guide.