Which mainstream area in mathematics surprises you by not being taught more widely? by Helen Tseng

Answer by Helen Tseng:

The Hexagon of Trigonometric Identities

This powerful mnemonic is easily reproducible by memory, with functions on one side and co-functions on the other, a 1 in the middle, and three triangles pointed down.

Reading across diagonals gives you inverses:

[math]\sin{x} = \frac{1}{\csc{x}}[/math]

[math]\cos{x} = \frac{1}{\sec{x}}[/math]

[math]\tan{x} = \frac{1}{\cot{x}}[/math]

Reading left-right-down the three triangles gives you these identities:

[math]\sin^2{x} + \cos^2{x} = 1[/math]

[math]\tan^2{x} + 1 = \sec^2{x}[/math]

[math]1 + \cot^2{x} = \csc^2{x}[/math]

Reading clockwise or counter-clockwise gives you these ratios:

[math]\sin{x} = \frac{\cos{x}}{\cot{x}}[/math]

or

[math]\sin{x} = \frac{\tan{x}}{\sec{x}}[/math]

and so forth…

Reading a function and multiplying the two nearest neighbors gives you these identities:

[math]\sin{x} = \cos{x} \cdot \tan{x}[/math]

[math]\cos{x} = \sin{x} \cdot \cot{x}[/math]

[math]\tan{x} = \sin{x} \cdot \sec{x}[/math]

and so forth…

Relations can be derived too. Ratio of any two left-right neighbours is the same:

[math]\frac{\sin{x}}{\cos{x}} = \frac{\tan{x}}{1} = \frac{1}{\cot{x}} = \frac{\sec{x}}{\csc{x}}[/math]


I should clarify that I am American and have only attended schools in the US. My dad taught me this when I was in middle school (apparently it's a standard trick in Taiwan) and I was surprised that none of my math teachers in middle and high school were aware of such an elegant mnemonic.

Which mainstream area in mathematics surprises you by not being taught more widely?

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