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Magic Hexagon

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Last updated date: 19th May 2024
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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 conjunctions. 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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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


Solved Examples

1. Which of the following Quotient Identities given below is accurate?

  1. Sin y = Sin (y)/Cot  (y)

  2. Sin y = Cos (y)/Cot (y)

  3. Sin y = Sec (y)/Tan ( y)

  4. Sin y = Cos (y)/Sec ( y)

Solution:


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The magic hexagon shows that the identity Sin y = Cos(y)/Cot(y) is correct.


2. Which of the following reciprocal identities is accurate?

  1. Cot y = 1/Tan  (y)

  2. Cot y = 1/Sin  (y)

  3. Cot y = 1/Sec (y)

  4. Cot y = 1/Csc  (y)

Solution:


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The magic hexagon shows that the identity Cot y = 1/Tan(y) is accurate.


Abnormal Magic Hexagons

Even though normal magic hexagons with orders of more than 3 are not known, certain “abnormal” magic hexagons are known to exist. In this case, the word "abnormal" refers to the state where the starting of the sequence of numbers is other than with 1. Such abnormal hexagons were first discovered by  Arsen Zahray, who discovered the order 4 and 5 magic hexagons


We can understand this by the following facts:  the order 4 hexagon starts with 3 and ends with 39, and the sum of its row to 111. Similarly, an order of 5 hexagons will begin with 15, end with 75, and add up to 305.


A total higher than 305 for an order 5 hexagon is not possible.


In a magic hexagon of 5th order, "X" will refer to the placeholders for order 3 hexagons, which complete the number sequence. In the upper fits the hexagon gives a total sum of 38 (numbers 1 to 19) while in the lower one of the 26 hexagons will have the sum equal to 0 (numbers -9 to 9).


An order of six hexagons starts with 21, ends with 111, and its sum is 546.  It was designed by mathematician Louis Hoelbling on October 11, 2004.


And the magic hexagon of order 7 was discovered  by Arsen Zahray with the help of annealing simulation method on 22 March 2006:


A seventh-order magic hexagon starts with 2, ends with 128, and sums up to the value of 635.


Similarly, the magic hexagon of the order 8 was created by Louis K. Hoelbling on February 5, 2006, and it begins with -84 and ends with 84, and its sum is 0.


Magic T- hexagons

It is interesting to know that magic hexagons can also be constructed with triangles and this type of configuration is referred to as a T-hexagon. It has several more properties than the hexagon of hexagons.


In general, a T-hexagon of order n has 6n2 triangles. The sum of all these numbers is given by: S = 3n2(6n2+1)

In order to make a magic T-hexagon of side n, 'n' must be even, because there are r = 2n rows so the sum in each row must be

M= S/r

There must be an even number of 'n' for 'M' to be an integer. To date, magic T-hexagons only of the order 2, 4, 6, and 8 have been discovered. The first magic hexagon that was a magic T-hexagon of order 2 was supposedly discovered by John Baker on 13 September 2003. Since then, John has had several collaborations with David King (who is well known for the discovery that there are 59,674,527 non-congruent magic T-hexagons of 2nd order).


Magic T-hexagons are sometimes thought to be similar in a number of ways to magic squares, but they have their own special features.

FAQs on Magic Hexagon

1. Define Magic Hexagon.

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?

The various trigonometric identities are represented with the help of magic hexagon in the following ways:

In the clockwise direction, we obtain the following quotient identities:

Tan x= sin x/ cos x; cot x= csc x/ sec x


Similarly, sin x can be written as cos x/ cot x and cosec x can be denoted as sec x/ tan x


For cos x, the derivation is cot x/ csc x and for sec x, the value is derived as tan x/ sin x


When moved in counter clockwise direction the quotient identities obtained are:

Sin x = tan x/ sec x; csc x= cot x/ cos x


Tan x= sec x/ csc x; cot x= cos x/ sin x and so on

3. List all the Fundamental Trigonometric Identities.

The fundamental trigonometric identities refer to the relation derived from its fundamental functions (sin, cos, tan). All fundamental trigonometric identities (there are eight of them) are listed below:

Cos x= 1/ sin x


Sin x= 1/ cos x


Cot x= 1/ tan x


Tan x= sin x/ cos x


Cot x= cos x/ sin x


Sin2x + cos2 x= 1


1 + tan2 x= sec2 x


1 + cot2 x= csc2 x

4. What is an abnormal hexagon?

Magic hexagons that are more than the order of three are known as abnormal hexagons. In them, the starting of the sequence of numbers is other than with 1. Such abnormal hexagons were first discovered by  Arsen Zahray, who discovered the order 4 and 5 magic hexagons. Magic hexagons of this type have been discovered for up to the order of 8.

5. How can we derive Pythagorean identities from a magic hexagon?

  • The unit circle in a magic hexagon shows that the value of sin² x + cos² x = 1

  • The magic hexagon helps us to remember the Pythagorean identities by moving in a clockwise direction, around any of the triangles of the hexagon.