View source for Mnemonics in trigonometry

{{Short description|Mathematical memory aids}}
In [[trigonometry]], it is common to use [[mnemonic]]s to help remember [[trigonometric identities]] and the relationships between the various [[trigonometric functions]].

== SOH-CAH-TOA ==
[[File:trigonometric_function_triangle_mnemonic.svg|thumb|Image mnemonic to help remember the ratios of sides of a right triangle]]
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:

:'''S'''ine = '''O'''pposite ÷ '''H'''ypotenuse
:'''C'''osine = '''A'''djacent ÷ '''H'''ypotenuse
:'''T'''angent = '''O'''pposite ÷ '''A'''djacent

One way to remember the letters is to sound them out phonetically (i.e. {{IPAc-en|ˌ|s|oʊ|k|ə|ˈ|t|oʊ|ə}} {{respell|SOH|kə|TOH|ə}}, similar to [[Krakatoa]]).<ref>{{Cite book |last=Humble |first=Chris |url=https://www.worldcat.org/oclc/47985033 |title=Key Maths : GCSE, Higher. |date=2001 |publisher=Stanley Thornes Publishers |others=Fiona McGill |isbn=0-7487-3396-5 |location=Cheltenham |oclc=47985033|page=51}}</ref>

===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).<ref name="mathworld">{{MathWorld|title=SOHCAHTOA|urlname=SOHCAHTOA}}</ref><ref>{{cite book |title=Memory: A Very Short Introduction|first=Jonathan K.|last=Foster|publisher=Oxford|year=2008|isbn=978-0-19-280675-8|page=128|url=https://books.google.com/books?id=ISBN9780192806758}}</ref> Communities in Chinese circles may choose to remember it as TOA-CAH-SOH, which also means 'big-footed woman' ({{lang-zh|c=大腳嫂|poj=tōa-kha-só}}) in [[Hokkien]].{{cn|date=February 2023}}

An alternate way to remember the letters for Sin, Cos, and Tan is to memorize the syllables Oh, Ah, Oh-Ah (i.e. {{IPAc-en|oʊ|_|ə|_|ˈ|oʊ|.|ə}}) for O/H, A/H, O/A.<ref>{{Mathworld|title=Trigonometry|urlname=Trigonometry}}</ref> Longer mnemonics for these letters include "Oscar Has A Hold On Angie" and "Oscar Had A Heap of Apples."<ref name="mathworld" />

==All Students Take Calculus==
[[File:trigonometric_function_quadrant_sign.svg|thumb|Signs of trigonometric functions in each quadrant.]]
'''All S'''tudents '''T'''ake '''C'''alculus is a [[mnemonic]] for the sign of each [[trigonometric functions]] in each [[Cartesian coordinate system|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.<ref name=steuben />
* Quadrant 1 (angles from 0 to 90 degrees, or 0 to π/2 radians): '''All''' trigonometric functions are positive in this quadrant.
* Quadrant 2 (angles from 90 to 180 degrees, or π/2 to π radians): '''S'''ine and cosecant functions are positive in this quadrant.
* Quadrant 3 (angles from 180 to 270 degrees, or π to 3π/2 radians): '''T'''angent and cotangent functions are positive in this quadrant.
* Quadrant 4 (angles from 270 to 360 degrees, or 3π/2 to 2π radians): '''C'''osine and secant functions are positive in this quadrant.

Other mnemonics include:
*'''All S'''tations '''T'''o '''C'''entral<ref name=mathfun>{{cite web |url=http://www.mathsisfun.com/algebra/trig-four-quadrants.html |title=Sine, Cosine and Tangent in Four Quadrants |accessdate=2015-01-18 |archiveurl=https://web.archive.org/web/20150118121241/http://www.mathsisfun.com/algebra/trig-four-quadrants.html |archivedate=2015-01-18 |work=Math Is Fun}}</ref>
*'''All S'''illy '''T'''om '''C'''ats<ref name=mathfun/>
*'''A'''dd '''S'''ugar '''T'''o '''C'''offee<ref name=mathfun/>
*'''All''' '''S'''cience '''T'''eachers (are) '''C'''razy<ref name="o698">{{cite book | last1=Heng | first1=H. H. | last2=Cheng | first2=Khoo | last3=Talbert | first3=J. F. | title=Additional Mathematics | publisher=Pearson Education South Asia | date=2005 | isbn=978-981-235-211-8 | page=228|url=https://books.google.com/books?id=ZZoxLiJBwOUC | archive-url=https://web.archive.org/web/20230610195637/https://books.google.com/books?id=ZZoxLiJBwOUC |archivedate=2023-06-10}}</ref>
*'''A''' '''S'''mart '''T'''rig '''C'''lass<ref>{{cite web |url=https://www.onlinemathlearning.com/mnemonics-for-trigonometry.html |title=Math Mnemonics and Songs for Trigonometry |work=Online Math Learning |accessdate=2019-10-17 |archive-date=2019-10-17 |archive-url=https://web.archive.org/web/20191017145136/https://www.onlinemathlearning.com/mnemonics-for-trigonometry.html |url-status=live }}</ref>
*'''All''' '''S'''chools '''T'''orture '''C'''hildren<ref name=steuben>{{Cite book |last1=Stueben |first1=Michael |title=Twenty years before the blackboard: the lessons and humor of a mathematics teacher |last2=Sandford |first2=Diane |date=1998 |publisher=Mathematical Association of America |isbn=978-0-88385-525-6 |series=Spectrum series |location=Washington, DC|page=119|url=https://books.google.com/books?id=qnd0P-Ja-O8C&dq=%22All+Schools+Torture+Children%22&pg=PA119}}</ref>
*'''A'''wful '''S'''tinky '''T'''rig '''C'''ourse<ref name=steuben />

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. 
* '''CAST''' still goes counterclockwise but starts in quadrant 4 going through quadrants 4, 1, 2, then 3.
* '''ACTS''' still starts in quadrant 1 but goes clockwise going through quadrants 1, 4, 3, then 2.

==Sines and cosines of special angles==
Sines and cosines of common angles 0°, 30°, 45°, 60° and 90° follow the pattern <math>\frac{\sqrt{n}}{2}</math> with {{math|1=n = 0, 1, ..., 4}} for sine and {{math|1=n = 4, 3, ..., 0}} for cosine, respectively:<ref>{{cite book|first=Ron |last=Larson |url=https://books.google.com/books?id=bsZDAwAAQBAJ&pg=PA275 |title=Precalculus with Limits: A Graphing Approach, Texas Edition|edition=6|date=2014|publisher=Cengage Learning}}</ref>
{| class="wikitable"
! <math>\theta</math> !! <math>\sin \theta</math> !! <math>\cos \theta</math> !! <math>\tan \theta = \sin \theta \Big/ \cos \theta</math>
|-
| style="text-align: right;"| 0° = 0 radians
| <math>\frac{\sqrt{\mathbf{\color{blue}{0}}}}{2} = \;\; 0</math>
| <math>\frac{\sqrt{\mathbf{\color{red}{4}}}}{2} = \;\; 1</math>
| <math>\;\; 0 \;\; \Big/ \;\; 1 \;\; = \;\; 0</math>
|-
| 30° = {{sfrac|{{pi}}|6}} radians
| <math>\frac{\sqrt{\mathbf{\color{teal}{1}}}}{2} = \;\, \frac{1}{2}</math>
| <math>\frac{\sqrt{\mathbf{\color{orange}{3}}}}{2}</math>
| <math>\;\, \frac{1}{2} \; \Big/ \frac{\sqrt{3}}{2} = \frac{1}{\sqrt{3}}</math>
|-
| 45° = {{sfrac|{{pi}}|4}} radians
| <math>\frac{\sqrt{\mathbf{\color{green}{2}}}}{2} = \frac{1}{\sqrt{2}}</math>
| <math>\frac{\sqrt{\mathbf{\color{green}{2}}}}{2} = \frac{1}{\sqrt{2}}</math>
| <math>\frac{1}{\sqrt{2}} \Big/ \frac{1}{\sqrt{2}} = \;\; 1</math>
|-
| 60° = {{sfrac|{{pi}}|3}} radians
| <math>\frac{\sqrt{\mathbf{\color{orange}{3}}}}{2}</math>
| <math>\frac{\sqrt{\mathbf{\color{teal}{1}}}}{2} = \; \frac{1}{2}</math>
| <math>\frac{\sqrt{3}}{2} \Big/ \; \frac{1}{2} \;\, = \sqrt{3} </math>
|-
| 90° = {{sfrac|{{pi}}|2}} radians
| <math>\frac{\sqrt{\mathbf{\color{red}{4}}}}{2} = \;\, 1</math>
| <math>\frac{\sqrt{\mathbf{\color{blue}{0}}}}{2} = \;\, 0</math>
| <math>\;\; 1 \;\; \Big/ \;\; 0 \;\; = </math> undefined
|}

==Hexagon chart==
[[File:Trigonometric identity mnemonic.svg|thumb|upright=1.1|Trigonometric identities mnemonic]] 

Another mnemonic permits all of the basic identities to be read off quickly. The hexagonal chart can be constructed with a little thought:<ref>{{cite web|url=https://www.mathsisfun.com/algebra/trig-magic-hexagon.html|title=Magic Hexagon for Trig Identities|website=Math is Fun|access-date=2018-02-04|archive-date=2018-02-05|archive-url=https://web.archive.org/web/20180205001201/https://www.mathsisfun.com/algebra/trig-magic-hexagon.html|url-status=live}}</ref>
# Draw three triangles pointing down, touching at a single point. This resembles a [[fallout shelter]] [[trefoil]].
# Write a 1 in the middle where the three triangles touch
# Write the functions without "co" on the three left outer vertices (from top to bottom: ''sine'', ''tangent'', ''secant'')
# Write the co-functions on the corresponding three right outer vertices (''co''sine, ''co''tangent, ''co''secant)

Starting at any vertex of the resulting hexagon:{{clear}}
{|class="wikitable" style="font-size:100%;text-align:left;"
|+
! Identity !! Example(s) !! Illustration
|-
| rowspan=3 | The starting vertex equals one over the opposite vertex. 
| style="background:#ffa;" | <math>\sin A = \frac{{1}}{{\csc A}}</math>
| rowspan=3 | [[File:Trigonometría - Identidades recíprocas.svg|frameless|upright=0.8]]
|-
| style="background:#faa;" | <math>\tan A = \frac{{1}}{{\cot A}}</math>
|-
| style="background:#afa;" | <math>\cos A = \frac{{1}}{{\sec A}}</math>
|-
| rowspan=6 | Going either clockwise or counter-clockwise, the starting vertex equals the next vertex divided by the vertex after that.
| <math>\sin A = \frac{{\cos A}}{{\cot A}} = \frac{{\tan A}}{{\sec A}}</math>
| rowspan=6 | [[File:Trigonometría - Cocientes 1.svg|frameless|upright=0.8]] [[File:Trigonometría - Cocientes 2.svg|frameless|upright=0.8]]
|-
| <math>\cos A = \frac{{\cot A}}{{\csc A}} = \frac{{\sin A}}{{\tan A}}</math>
|-
| <math>\cot A = \frac{{\csc A}}{{\sec A}} = \frac{{\cos A}}{{\sin A}}</math>
|-
| <math>\csc A = \frac{{\sec A}}{{\tan A}} = \frac{{\cot A}}{{\cos A}}</math>
|-
| <math>\sec A = \frac{{\tan A}}{{\sin A}} = \frac{{\csc A}}{{\cot A}}</math>
|-
| <math>\tan A = \frac{{\sin A}}{{\cos A}} = \frac{{\sec A}}{{\csc A}}</math>
|-
| The starting corner equals the product of its two nearest neighbors. 
| <math>\sin A = \cos A \cdot \tan A</math>
| [[File:Trigonometría - Productos.svg|frameless|upright=0.8]]
|-
| rowspan=3 | The sum of the squares of the two items at the top of a triangle equals the square of the item at the bottom. These are the [[Pythagorean trigonometric identity|trigonometric Pythagorean identities]].
| style="background:#ffa;" | <math>\sin^2 A + \cos^2 A = 1^2 = 1 \ </math>
| rowspan=3 | [[File:Trigonometría - Identidades pitagóricas.svg|frameless|upright=0.8]]
|-
| style="background:#aff;" | <math>1 + \cot^2 A = \csc^2 A \ </math>
|-
| style="background:#faf;" | <math>\tan^2 A + 1 = \sec^2 A \ </math>
|}

Aside from the last bullet, the specific values for each identity are summarized in this table:
{| class="wikitable"
|-
! Starting function !! ... equals {{sfrac|1|opposite}} !! ... equals {{sfrac|first|second}} clockwise  !! ... equals {{sfrac|first|second}} counter-clockwise/anticlockwise  !! ... equals the product of two nearest neighbors
|-
| <math>\tan A</math> || <math> = \frac {1}{\cot A}</math> || <math> = \frac {\sin A}{\cos A} </math> || <math>= \frac {\sec A}{\csc A} </math> || <math>= \sin A \cdot \sec A</math>
|-
| <math>\sin A</math> || <math> = \frac {1}{\csc A}</math> || <math> = \frac {\cos A}{\cot A} </math> || <math>= \frac {\tan A}{\sec A} </math> || <math>= \cos A \cdot \tan A</math>
|-
| <math>\cos A</math> || <math> = \frac {1}{\sec A}</math> || <math> = \frac {\cot A}{\csc A} </math> || <math>= \frac {\sin A}{\tan A} </math> || <math>= \sin A \cdot \cot A</math>
|-
| <math>\cot A</math> || <math> = \frac {1}{\tan A}</math> || <math> = \frac {\csc A}{\sec A} </math> || <math>= \frac {\cos A}{\sin A} </math> || <math>= \cos A \cdot \csc A</math>
|-
| <math>\csc A</math> || <math> = \frac {1}{\sin A}</math> || <math> = \frac {\sec A}{\tan A} </math> || <math>= \frac {\cot A}{\cos A} </math> || <math>= \cot A \cdot \sec A</math>
|-
| <math>\sec A</math> || <math> = \frac {1}{\cos A}</math> || <math> = \frac {\tan A}{\sin A} </math> || <math>= \frac {\csc A}{\cot A} </math> || <math>= \csc A \cdot \tan A</math>
|}

== See also ==

* [[List of trigonometric identities]]

== References ==
{{reflist}}

{{Math mnemonics}}

[[Category:Trigonometry]]
[[Category:Science mnemonics]]