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The Hexagon‘trig trick’

The relationships between the trigonometric functions can be represented in this fun and easy to remember way:

Place the names of the sixtrigonometric functions sin θ, cos θ, tan θ, cot θ, sec θ, and cosec θ at thevertices labelled A, B, C,D, E, and F, respectively.This step has to be donein this manner or else thetrick will not work.

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sin ϴ cos ϴ

tan ϴ cot ϴ

sec ϴ cosec ϴ

Shading/Colouring

Shade or colour in the triangle with vertices sin θ, cos θ, and the centre. Then shade in the triangle whose vertices are at tan θ, sec θ, and the centre, and the triangle whose vertices are at cot θ, cosec θ , and the centre.

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cos Ө

cosec Өsec Ө

sin Ө

tan Ө cot Ө1

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Co-function Relations.

The trig functions cosine, cotangent, andcosecant on the right of the hexagon areco-functions of sine, tangent, and secanton the left, respectively.

sin (90°- θ) = cos θsec (90°- θ) = cosec θtan (90°- θ) = cot θ

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Reciprocal Identities

The two trig functions on any diagonal arereciprocals of each other.

sin θ = 1 ÷ cosec θcos θ = 1 ÷ sec θtan θ = 1 ÷ cot θ

Product Identities

Along the outside edges of the hexagon anytrig function equals the product of the functionson the adjacent vertices:

sin θ = cos θ x tan θcos θ = sin θ x cot θcot θ = cos θ x cosec θcosec θ = cot θ x sec θsec θ = tan θ x cosec θtan θ = sin θ x sec θ

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Quotient Identities

Using the product identities we can alsofind the quotient identities:

tan θ = sin θ ÷ cos θcot θ = cos θ ÷ sin θ

Pythagorean Identities

For each shaded triangle, the upper-leftfunction squared plus the upper-rightfunction squared equals the bottomfunction squared.You can use the number seven drawn on yourdiagram to assist you.

sin 2 θ + cos 2 θ = 1tan 2 θ + 1 = sec 2 θ1 + cot 2 θ = cosec 2 θ

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cos Ө

cosec Өsec Ө

sin Ө

tan Ө cot Ө1

References:

Chien, V. (n.d) Trigonometry Triangleswww.pen.k12.va.us/Div/Winchester/jhhs/math/lessons/trig/hex.html retrieved 08/06/07Dave's Short Trig Course http://www.clarku.edu/~djoyce/trig/ Retrieved 11/06/07Department of Education, (2003). National Curriculum Statement Grades 10-12 (General) Mathematics, Cape Town.