Magic Hexagon

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Magic Hexagon For Trigonometry

Magic hexagon of order n is an array of numbers in a centered hexagonal pattern with n cells on each edge, in such a way that the numbers in each row, in all three directions of magic hexagon sum to the same magic constant M. A normal magic hexagon includes the consecutive integer from 1 to 3n² - 3n + 1, whereas the abnormal magic hexagon starts with the number other than 1. It is concluded that normal magic hexagons exist only for n = 1 (which is trivial) or n = 3. The first magic hexagon that was introduced has a magic sum of 1 and the second magic hexagon has a sum of 38.


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The numbers in any row of the above hexagon  with order n = 3 sums to 38.  For example, 3 + 17 + 18 = 38, 19 + 7 + 1 + 11 = 38, 12 + 4 + 8 + 14 = 38. 


A magic hexagon for trigonometric identities is a special diagram that helps you to remember trigonometric identities. Here, we look at how the magic hexagon for trigonometry helps to remember different trigonometric identities.


Magic Hexagon For Trigonometric Identities

Trigonometric identities are equalities that include trigonometric functions and are true for every value of the variables that occur for which both the sides of equalities are defined. The trigonometric identities are useful whenever expressions including trigonometric functions are required to be simplified.  


The magic hexagon is a special diagram that helps you to quickly memorize different trigonometric identities such as Pythagorean, reciprocal, product/function, and cofunction identities. Also, you will learn how trigonometric identities are useful to evaluate trigonometric functions.


Building Magic Hexagon For Trigonometric Identities

  1. Construct the hexagon and locate ‘I’ at the center of the hexagon.

  2. Write tan on the farthest left vertex of the magic hexagon as shown in the figure given below.

  3. Use quotient identities ( tan x = sin x/cos x ) for the tangent going clockwise as shown in the figure below.

  4. Locate the “co” - functions such as cot (cotangent), csc (cosecant), and sec (secant) on the opposite vertex of the hexagon as shown in the figure below.

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To help you remember: all the “co” functions are placed on the right side of the hexagon.


Along the outside edges of the hexagon, we can now follow around the clock (in either direction) to get the quotient identities. The quotient identities given below are in two equivalent forms of each.


Quotient Identities

Clockwise 

Counter Clockwise

Tan x = Sin x/Cos x

Cot x = Csc x/Sec x

Cos x = Sin x/Tan x 

Secx = Cscx/Cotx

Sin x = Cos x/Cot x

Csc x = Sec x/Tan  x

Sin x = Tan x/Sex x

Cscx = Cotx/Cosx

Cos x = Cot x/Csc x

Sec x = Tan x/Sin  x

Tan x = Sec x/Csc x

Cotx = Cosx/Sinx


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The two trigonometric functions located on any diagonals of a hexagon are reciprocal of each other. 


Reciprocal Identities

Sin x = 1/Csc x

Cosx = 1/Sec x

Tan x = 1/Cot x

Csc x =  1/Sin x

Sec x = 1/Cos x

Cot x = 1/Tan x


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Product Identities

The magic hexagon given here shows that the trigonometric function between two functions is equal to them and multiplied together. If the identities are opposite to each other, then 1 is between them. Check the product identities below to know better.


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Sinx = tanx cosx

Cosx = Sinx Cotx

Cscx = Cotx secx

Sec x = Tanx Cscx

Tanx = Sinx Secx 

Cotx = Cosx Cscx


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Tan x.Cot x = 1

Sec x.Cos x = 1

Sin x.Csc x = 1


Cofunction Identities

The trigonometric functions such as cosine, cotangent, and cosecant on the right side of the hexagon are the cofunction of the trigonometric functions that are on the left side of the hexagon such as sine, tangent, and secant. Hence, sine and cosine are cofunctions. Let us learn to form the cofunction identities with the help of the figure given below.


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Examples:

  • The value of sine (30°) = cos (60°)

  • The value of tan (80°) = cos (10°)

  • The value of sec (40°) = csc (50°)

Cofunction Identities in Radians


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Cofunction Identities in Radians

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Examples:

  • The value of sine (0.1π) = cos (0.4 π)

  • The value of tan (π/4) = cot (π/4)

  • The value of sec (π/3) = csc (π/6)

The Pythagorean Identities

The unit circle shows that sin² x + cos² x = 1

The magic hexagon also helps us to remember Pythagorean identities by moving clockwise, around any of the triangles below.

For each shaded triangle in the figure below, the addition of the upper left function squared and upper right function squared obtains the bottom function square.


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We can write Pythagorean Identities as:


Sin² x = 1 - cos² x

tan² x = Sec² x -1

Cot² x = Csc² - 1

Cos² x = 1 - sin² x

Sec² x = tan² x + 1

Csc² x = 1 + cot² x

FAQ (Frequently Asked Questions)

1. Define Magic Hexagon.

Ans: Magic Hexagon is a centered hexagonal pattern of cells comprising the natural numbers from 1 to the number of cells such that the numbers in all the cells given in a straight line all sum up to the same number. 


The main aim of the magic hexagon is to arrange the numbers in such a  way that each row across and diagonal adds to the same number. Magic hexagonal problem was first published by Ernst Von Haselberg in 1888.

2. How Different Trigonometry Identities are Represented in Magic Hexagon?

Ans: Trigonometry identities like reciprocal, quotient, product and, Pythagorean are represented in the magic hexagon in the following ways:

  • Reciprocal Identity - The diagonals of the magic hexagon represent the reciprocal identity.

  • Quotient Identity - Select a vertex of the magic hexagon and move either in a clockwise or counterclockwise direction.

  • Product Identity - Product identity is the product of the surrounding trigonometry functions. Select a vertex, if trigonometry identities are opposite to each other, then 1 is between them.

3. List all the Fundamental Trigonometric Identities.

Ans: There are various trigonometric identities that are used in Mathematics to solve trigonometric problems. Here is the list of fundamental trigonometric identities:

  • Reciprocal Identity

  • Ratio Identity

  • Product Identity

  • Angle Sum and Difference Identity

  • Opposite angle identity

  • Complementary Angle Identity