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
Construct the hexagon and locate ‘I’ at the center of the hexagon.
Write tan on the farthest left vertex of the magic hexagon as shown in the figure given below.
Use quotient identities ( tan x = sin x/cos x ) for the tangent going clockwise as shown in the figure below.
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
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The two trigonometric functions located on any diagonals of a hexagon are reciprocal of each other.
Reciprocal Identities
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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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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:
Solved Examples
1. Which of the following Quotient Identities given below is accurate?
Sin y = Sin (y)/Cot (y)
Sin y = Cos (y)/Cot (y)
Sin y = Sec (y)/Tan ( y)
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?
Cot y = 1/Tan (y)
Cot y = 1/Sin (y)
Cot y = 1/Sec (y)
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. What is the Magic Hexagon in the context of trigonometry?
In trigonometry, the Magic Hexagon is not a mathematical puzzle but a powerful mnemonic device used to remember a wide range of trigonometric identities. It is a hexagonal diagram with the six trigonometric functions (sin, cos, tan, cot, sec, csc) at its vertices and the number 1 in the centre. Its specific arrangement allows students to derive quotient, reciprocal, and Pythagorean identities easily.
2. How does the Magic Hexagon help in remembering quotient identities?
The Magic Hexagon provides a simple way to find quotient identities by moving clockwise or anti-clockwise around the hexagon.
- To find the identity for any function, you can divide the next function in the sequence by the one following it.
- For example, moving clockwise, tan(x) = sin(x) / cos(x).
- Similarly, moving anti-clockwise, tan(x) = sec(x) / csc(x).
3. How are the three Pythagorean identities derived from the Magic Hexagon?
The Pythagorean identities can be found by focusing on the three downward-pointing triangles within the hexagon. For each of these triangles:
- Start at the top-left vertex and square it.
- Add it to the square of the top-right vertex.
- This sum equals the square of the bottom vertex.
- sin²(x) + cos²(x) = 1²
- 1² + cot²(x) = csc²(x)
- tan²(x) + 1² = sec²(x)
4. What is the importance of the number '1' at the centre of the Magic Hexagon?
The number '1' at the centre is crucial as it links several types of identities. Firstly, it completes the reciprocal identities: any two functions on opposite sides of the hexagon (connected by a line passing through the '1') are reciprocals, meaning their product is 1. For example, sin(x) * csc(x) = 1. Secondly, the '1' acts as a key component in deriving two of the three Pythagorean identities (tan²(x) + 1 = sec²(x) and 1 + cot²(x) = csc²(x)), solidifying its central role in the hexagon's logic.
5. How can you find reciprocal identities using the Magic Hexagon?
Reciprocal identities are found by looking at the functions that are diagonally opposite each other, with the line connecting them passing through the '1' in the center. The rule is that the product of these two functions is 1. For example:
- sin(x) and csc(x) are opposite, so sin(x) = 1/csc(x).
- cos(x) and sec(x) are opposite, so cos(x) = 1/sec(x).
- tan(x) and cot(x) are opposite, so tan(x) = 1/cot(x).
6. What is the relationship between horizontally adjacent functions in the Magic Hexagon?
Functions that are horizontally next to each other in the Magic Hexagon are co-functions. This means one is the co-function of the other. For instance:
- sin(x) is next to cos(x), representing the identity sin(x) = cos(90° - x).
- tan(x) is next to cot(x), representing tan(x) = cot(90° - x).
- sec(x) is next to csc(x), representing sec(x) = csc(90° - x).
7. Are there any trigonometric identities that the Magic Hexagon cannot help you remember?
Yes, while the Magic Hexagon is extremely useful, it has limitations. It is primarily designed for fundamental identities like quotient, reciprocal, Pythagorean, and co-function identities. It does not provide a direct way to remember more advanced formulas such as:
- Sum and Difference Identities (e.g., sin(A + B))
- Double-Angle and Half-Angle Identities (e.g., sin(2x))
- Product-to-Sum Identities
