Mnemonics in trigonometry

In trigonometry, it is common to use mnemonics to help remember trigonometric identities and the relationships between the various trigonometric functions. For example, 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. "SOH-CAH-TO-A").^[1] Another method is to expand the letters into a sentence, such as "Some Old Hippy Caught Another Hippy Trippin' On Acid".^[2]

Mnemonic chart

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 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 radiation symbol.)

Trigonometric identities mnemonic

Reading across the central 1 in any direction gives reciprocal identities:

Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \begin{array}{} {1 \over \sin A} = \csc A & \text{or} & {1 \over \csc A} = \sin A \\ \\ {1 \over \tan A} = \cot A & \text{or} & {1 \over \cot A} = \tan A \\ \\ {1 \over \sec A} = \cos A & \text{or} & {1 \over \cos A} = \sec A \end{array}

Reading down any triangle gives the standard identities (starting at the top and going clockwise):

$\sin^2 A + \cos^2 A = 1 \$

$1 + \cot^2 A = \csc^2 A \$

$\tan^2 A + 1 = \sec^2 A \$

Reading a function and dividing the two consecutive clockwise or counter clockwise neighbors gives these identities:

(Starting at tan and going clockwise)

$\tan A = {\sin A \over \cos A}$

$\sin A = {\cos A \over \cot A}$

$\cos A = {\cot A \over \csc A}$

$\cot A = {\csc A \over \sec A}$

$\csc A = {\sec A \over \tan A}$

$\sec A = {\tan A \over \sin A}$

(Starting at tan and going counter-clockwise)

$\tan A = {\sec A \over \csc A}$

$\sec A = {\csc A \over \cot A}$

$\csc A = {\cot A \over \cos A}$

$\cot A = {\cos A \over \sin A}$

$\cos A = {\sin A \over \tan A}$

$\sin A = {\tan A \over \sec A}$

Reading a function and multiplying the two nearest neighbors gives these identities (starting at tan and going clockwise):

$\tan A = \sin A \cdot \sec A \$

$\sin A = \cos A \cdot \tan A \$

$\cos A = \sin A \cdot \cot A \$

$\cot A = \cos A \cdot \csc A \$

$\csc A = \cot A \cdot \sec A \$

$\sec A = \csc A \cdot \tan A \$

References

1. ^ Weisstein, Eric W., "SOHCAHTOA" from MathWorld.
2. ^ A sentence that is more appropriate for high school is, "Some old horse came a'hopping through our alley." Foster, Jonathan K. (2008). Memory: A Very Short Introduction. Oxford. p. 128. ISBN 0192806750.