Mnemonics in trigonometry


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:
One way to remember the letters is to sound them out phonetically.
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" or "The Old Army Colonel And His Son Often Hiccup". Communities in Chinese circles may choose to remember it as TOA-CAH-SOH, which also means 'big-footed woman' in Hokkien.
An alternate way to remember the letters for Sin, Cos, and Tan is to memorize the nonsense syllables Oh, Ah, Oh-Ah for O/H, A/H, O/A. Or, to remember all six functions, Sin, Cos, Tan, Cot, Sec, and Csc, memorize the syllables O/H, A/H, Oh/Ah, Ah/Oh, H/A, H/O. Longer mnenmonics 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.
Another mnemonic permits all of the basic identities to be read off quickly. Although the word part of the mnemonic used to build the chart does not hold in English, the chart itself is fairly easy to reconstruct with a little thought. Functions without "co" appear on the left, co-functions on the right, a 1 goes in the middle, triangles point down, and the entire drawing looks like a fallout shelter trefoil.
Starting at any corner of the hexagon:
Aside from the last bullet, the specific values for each identity are summarized in this table:
Starting function... equals one over the opposite... equals the first over the second, going clockwise... equals the first over the second, going counter-clockwise... equals the product of two nearest neighbors