Trigonometry mnemonics

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A number of mnemonics have been invented by educators to help students remember the rules defining the various trigonometric functions. Some of them are listed here, in no particular order.

Contents

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Mnemonics for recalling the definitions of sine, cosine and tangent

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Mnemonics with explicit sine, cosine and tangent

These mnemonics take the first letters of

Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent

and form some easily-remembered sentence. Popular examples are

A war cry Soh Cah Toa

  • Some Officers Have Coaches And Horses To Order Around
  • Some Old Horses Chase And Hunt Till Old Age
  • Some Old Hippie Caught A High Tripping On Acid
  • Some old Hippe Caught Another Hippie Tripping on Acid
  • Signs Of Happiness Come After Having Tubs Of Acid
  • Some Old Hags Can't Always Hide Their Old Age
  • Some Old Hobos Can't Always Hide Their Old Age
  • Some Old Horses Can Always Hear Their Owner's Approach
  • Silly Old Harry Caught A Herring Trawling Off America
  • Silly Old Hippies Can Always Have Tonnes Of Acid
  • Some Old Hippie/Hippy Came Around Here Tripping On Acid (or "...Caught Another Hippie...")
  • SOHCAHTOA (sounds like "soak a toe-a", can be read as "soccer tour")
  • Silly Old Hitler Chased A Horse Through Our Attic
  • Silly Old Hitler Caused Awful Headaches To Our Airmen
  • Sir Olivers Horse Came Ambling Home to Olivers Aunt
  • Some Old Hag Cracked All Her Teeth On Asparagus
  • Shit Out His Cummy Ass Hole Toward Old Adolf
  • Some Orifices Have Curly Auburn Hair To Obscure Approach (RAF mnemonic from WW2)
  • Saddle Our Horses Canter Away Happily To Other Adventures
  • See Old Harry Catch A Herring Trawling Off America
  • The Old Aunt Sat On Her Coat And Hat
  • Selly Oak Hospital Can Always Help The Odd Accident
  • SAKATA from the television advert.
  • Silly Old Hag, Can't Afford a Helicopter, Too Old Anyway (credit to Laura Pickett, Isle of Wight)
  • Sex On Holiday Causes All Heartache, Treatment Only Aggravates
  • Suck On Her Cunt And Have Two Orgasms After

Perhaps a somewhat stranger memory trick involves imagining that you are throwing a bucket of water at the tower of London, in which The Queen is currently residing. She responds to the water by saying:"Hey, Phil, someone's trying to Soak Our Tower!". Because of The Queen's more formal accent, it sounds more like SOH CAH TOA.(So kar to-a). As daft as it sounds, the bizzareness of it should help you to remember it.

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Mnemonics with implicit sine, cosine and tangent

These mnemonics do not include the function name. Each pair of words implicitly refers to sine, cosine, and tangent respectively. The first letters of

Opposite/Hypotenuse (=Sine)
Adjacent/Hypotenuse (=Cosine)
Opposite/Adjacent (=Tangent)

are used to form a sentence. Popular examples are

  • Old Houses Always Have Old Attics
  • Oh Heck, Another Hour Of Algebra!
  • Oscar Had A Hit Of Acid
  • Old Hippies Are High On Acid
  • Oscar Has A Hairy Old Ass
  • Oscar Has A Heap Of Acorns
  • Oscar Had A Hunk Of A..
  • Oscar Has A Heap Of Apples
  • Oscar Had A Hold On Arthur
  • Orla Has A Hell Of A...
  • One Hopes And Hopes On America
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Mnemonics for recalling which functions are positive in which quadrant

These mnemonics help recall which functions are positive in which quadrant, starting at the conventional 0 (east on a compass) and going counter-clockwise. They take the first letters of

All functions are positive in quadrant I
Sine is positive in quadrant II
Tangent is positive in quadrant III
Cosine is positive in quadrant IV

and form a simple sentence. Popular examples are

  • All Schools Teach Crap
  • All Silver Tea Cups
  • All Students Take Calculus
  • All Students Take Cocaine
  • A Simple Trig Chart
  • All Stations To Central
  • All Stations To Crewe
  • All Sinners Take Confession
  • All Science Teachers Care
  • All Students Take Chemistry
  • Another Stupid Trig Class
  • All Socialists Talk Crap
  • Albany State Teachers' College
  • Add Sugar To Coffee
  • All Smackheads Take Crack

Or following the above rule except starting with quadrant IV, the word CAST (Cosine, All, Sine, Tangent) is formed.

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Other mnemonics

These mnemonics define the same functions as above, but in a different order. Popular examples are

  • Old Harry Spills All His Custard Over Auntie's Tablecloth
Old Harry Spills: Opposite/Hypotenuse = Sine
All His Custard: Adjacent/Hypotenuse = Cosine
Over Auntie's Tablecloth: Opposite/Adjacent = Tangent
  • Oly And Olivia Have Hairy Ankles (defines Sine | Cosine | Tangent as follows)
Oly And Olivia: Opposite | Adjacent | Opposite
Have Hairy Ankles: Hypotenuse | Hypotenuse | Adjacent
  • Oranges Have Sections and Apples Have Cores.
Oranges Have: Opposite/Hypotenuse (=Sine)
Apples Have: Adjacent/Hypotenuse (=Cosine)
  • Sally Can Tell Oscar Has A Hat On Always
Sally Can Tell: Sine, Cosine, Tangent
Oscar Has: Opposite/Hypotenuse (=Sine)
A Hat: Adjacent/Hypotenuse (=Cosine)
On Always: Opposite/Adjacent (=Tangent)
  • The Cat Sat On An Orange And Howled Horribly
The Cat Sat: Tangent, Cosine, Sine
On An Orange: Opposite, Adjacent, Hypotenuse (=top of each ratio)
And Howled Horribly: Adjacent, Hypotenuse, Hypotenuse (=bottom of each ratio)
This is meant to be read vertically, in columns.
  • The Old Arab Sat On His Camel And Howled
The Old Arab: Tangent = Opposite/Adjacent
Sat On His: Sine = Opposite/Hypotenuse
Camel And Howled: Cosine = Adjacent/Hypotenuse

Another mnemonic uses rhythm to help students remember the sine and cosine of the sum of two angles.

  • Sin Sin Co Co Sin: Sin(A + B) = Sin(A)Cos(B) + Cos(A)Sin(B)
  • Co Co Co Sin Sin: Cos(A + B) = Cos(A)Cos(B) - Sin(A)Sin(B)
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