Mnemonics in trigonometry explained

In trigonometry, it is common to use mnemonics to help remember trigonometric identities and the relationships between the various trigonometric functions.

SOH-CAH-TOA

The sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, for instance SOH-CAH-TOA in English:

Sine = Opposite ÷ Hypotenuse

Cosine = Adjacent ÷ Hypotenuse

Tangent = Opposite ÷ Adjacent

One way to remember the letters is to sound them out phonetically (i.e., similar to Krakatoa).[1]

Phrases

Another method is to expand the letters into a sentence, such as "Some Old Horses Chew Apples Happily Throughout Old Age", "Some Old Hippy Caught Another Hippy Tripping On Acid", or "Studying Our Homework Can Always Help To Obtain Achievement". The order may be switched, as in "Tommy On A Ship Of His Caught A Herring" (tangent, sine, cosine) or "The Old Army Colonel And His Son Often Hiccup" (tangent, cosine, sine) or "Come And Have Some Oranges Help To Overcome Amnesia" (cosine, sine, tangent).[2] Communities in Chinese circles may choose to remember it as TOA-CAH-SOH, which also means 'big-footed woman' (Chinese: c=大腳嫂|poj=tōa-kha-só) in Hokkien.

An alternate way to remember the letters for Sin, Cos, and Tan is to memorize the syllables Oh, Ah, Oh-Ah (i.e.) for O/H, A/H, O/A. Longer mnemonics for these letters include "Oscar Has A Hold On Angie" and "Oscar Had A Heap of Apples."

All Students Take Calculus

All Students Take Calculus is a mnemonic for the sign of each trigonometric functions in each quadrant of the plane. The letters ASTC signify which of the trigonometric functions are positive, starting in the top right 1st quadrant and moving counterclockwise through quadrants 2 to 4.

Other mnemonics include:

Other easy-to-remember mnemonics are the ACTS and CAST laws. These have the disadvantages of not going sequentially from quadrants 1 to 4 and not reinforcing the numbering convention of the quadrants.

Sines and cosines of special angles

Sines and cosines of common angles 0°, 30°, 45°, 60° and 90° follow the pattern

\sqrt{n
} with for sine and for cosine, respectively:[7]

\theta

\sin\theta

\cos\theta

\tan\theta=\sin\theta/\cos\theta

0° = 0 radians
\sqrt{\color{blue{0
}}} = \;\; 0
\sqrt{\color{red{4
}}} = \;\; 1

  0  /  1  =  0

30° = radians
\sqrt{\color{teal{1
}}} = \;\, \frac
\sqrt{\color{orange{3
}}}

1
2

/

\sqrt{3
} = \frac
45° = radians
\sqrt{\color{green{2
}}} = \frac
\sqrt{\color{green{2
}}} = \frac
1
\sqrt{2
} \Big/ \frac = \;\; 1
60° = radians
\sqrt{\color{orange{3
}}}
\sqrt{\color{teal{1
}}} = \; \frac
\sqrt{3
} \Big/ \; \frac \;\, = \sqrt
90° = radians
\sqrt{\color{red{4
}}} = \;\, 1
\sqrt{\color{blue{0
}}} = \;\, 0

  1  /  0  =

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Hexagon chart

Another mnemonic permits all of the basic identities to be read off quickly. The hexagonal chart can be constructed with a little thought:[8]

  1. Draw three triangles pointing down, touching at a single point. This resembles a fallout shelter trefoil.
  2. Write a 1 in the middle where the three triangles touch
  3. Write the functions without "co" on the three left outer vertices (from top to bottom: sine, tangent, secant)
  4. Write the co-functions on the corresponding three right outer vertices (cosine, cotangent, cosecant)

Starting at any vertex of the resulting hexagon:

\sinA=

{1
}

\sinA=

{\cosA
} = \frac

\sinA=\cosA\tanA

\sin2A+\cos2A=12=1

1+\cot2A=\csc2A

\tan2A+1=\sec2A

Aside from the last bullet, the specific values for each identity are summarized in this table:

Starting function ... equals ... equals clockwise ... equals counter-clockwise/anticlockwise ... equals the product of two nearest neighbors

\tanA

=

1
\cotA

=

\sinA
\cosA

=

\secA
\cscA

=\sinA\secA

\sinA

=

1
\cscA

=

\cosA
\cotA

=

\tanA
\secA

=\cosA\tanA

\cosA

=

1
\secA

=

\cotA
\cscA

=

\sinA
\tanA

=\sinA\cotA

\cotA

=

1
\tanA

=

\cscA
\secA

=

\cosA
\sinA

=\cosA\cscA

\cscA

=

1
\sinA

=

\secA
\tanA

=

\cotA
\cosA

=\cotA\secA

\secA

=

1
\cosA

=

\tanA
\sinA

=

\cscA
\cotA

=\cscA\tanA

See also

Notes and References

  1. Book: Humble, Chris . Key Maths : GCSE, Higher. . 2001 . Stanley Thornes Publishers . Fiona McGill . 0-7487-3396-5 . Cheltenham . 47985033. 51.
  2. Book: Foster, Jonathan K.. Memory: A Very Short Introduction. Oxford. 2008. 978-0-19-280675-8. 128.
  3. Web site: Sine, Cosine and Tangent in Four Quadrants . 2015-01-18 . https://web.archive.org/web/20150118121241/http://www.mathsisfun.com/algebra/trig-four-quadrants.html . 2015-01-18 . Math Is Fun.
  4. Book: Heng . H. H. . Cheng . Khoo . Talbert . J. F. . Additional Mathematics . Pearson Education South Asia . 2005 . 978-981-235-211-8 . 228. https://web.archive.org/web/20230610195637/https://books.google.com/books?id=ZZoxLiJBwOUC . 2023-06-10.
  5. Web site: Math Mnemonics and Songs for Trigonometry . Online Math Learning . 2019-10-17 . 2019-10-17 . https://web.archive.org/web/20191017145136/https://www.onlinemathlearning.com/mnemonics-for-trigonometry.html . live .
  6. Book: Stueben . Michael . Twenty years before the blackboard: the lessons and humor of a mathematics teacher . Sandford . Diane . 1998 . Mathematical Association of America . 978-0-88385-525-6 . Spectrum series . Washington, DC. 119.
  7. Book: Larson, Ron . Precalculus with Limits: A Graphing Approach, Texas Edition. 6. 2014. Cengage Learning.
  8. Web site: Magic Hexagon for Trig Identities. Math is Fun. 2018-02-04. 2018-02-05. https://web.archive.org/web/20180205001201/https://www.mathsisfun.com/algebra/trig-magic-hexagon.html. live.