BIBHORR FORMULA: SHORTCUT TO TRIGONOMETRY

BIBHORR

0. INTRODUCTION

Trigonometry is the study of triangles with the aim of setting up relations between the sides and angles of

triangles. Since, the historical concepts did not allow any direct methodology of constructing these relations,

the traditional trigonometry or conventional trigonometry relies heavily on constant ratios that exist in the form of

trigonometric functions. These functions are solely the pre-calculated constant values for each angle corresponding

to the paired sides.

The measurement of trigonometric ratios is achieved through interpolation techniques like Taylor series. These

estimations help in approximating the ratios between the two paired sides against each angle value. This allow

scholars to prepare a long list of approximated constant values called Trigonometric tables. The indirect

trigonometric computations are not only slow and inaccurate

learners.

This research paper introduces direct methods of manifesting the side-angle relations for right triangles, without

any entanglement with the trigonometric functions.

1. NOTATIONS, DEFINTIONS AND AUXILIARY RESULTS

1.1 Notations and Definitions. For a right triangle ABC, the longest side or the hypotenuse BC will be called

shrav and its magnitude will be denoted by . Similarly, the longer and shorter sides (AC & AB) that are

mutually perpendicular to each other will be called lambu and chhutku and denoted by and respectively.

The angle ∠ ABC that lies opposite to the side AC (lambu) has been called Bibhorr angle; denoted by . ∠ ACB

that lies opposite to the side AB (chhutku) is the Ubhorr angle and will be denoted by ऊ .

Here, the author has defined each angle and each side for right triangle because these elements could now be

used to fit in a single equation as independent variables.

FIGURE 1. Notations for a right triangle.

Consider an arc CD spanning between BC and BD with radius BC (= BD). This will be referred to as Bibhorr

vakra and its length will be represented as . The radius of this arc is the shrav (longest side) of a right

triangle.

Consider another arc EF spanning 90º between AE and AF with radius AF (=AE). This arc has been referred

to as Kālpanik vakra notated as क . The radius of this arc has been called Bibhorr leek notated as .

ABSTRACT. This research paper explores new mathematical findings that are

ultimately found useful in the field of trigonometry. The findings are

completely new and are not grounded on any past theorem/formula or

concept. The result arising out of these findings simplifies the computations

without the involvement of trigonometric functions. The denouements can

also be found useful in other mathematical concepts.

B.Tech (Aerospace)

Indira Gandhi National Open University

New Delhi, INDIA

[bibhorr@zoho.com]

A B

C

(shrav)

(chhutku)

(lambu)

ऊ

but also complicated, at least to the novice

BIBHORR FORMULA : SHORTCUT TO TRIGONOMETRY | BY BIBHORR

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FIGURE 2. Notations for Arcs associated to right triangle.

1.2 Auxiliary Results. From Figure 2, the following two assertions are evident.

Assertion 1.1. Bibhorr vakra is the product of Bibhorr angle and shrav i.e. = .

Assertion 1.2. Kālpanik vakra is the product of 90º and Bibhorr leek i.e. क= 90 .

The following result would be assumed throughout this research paper.

Assumption 1.3. Bibhorr vakra equals Kālpanik vakra.

i.e. = क

or

= 90

2. BIBHORR LAW

Law 2.0 (Bibhorr law). The product of Bibhorr angle and shrav equals the product of right angle (90º) and

Bibhorr leek.

= 90

This law is merely the restatement of Assumption 1.3 mentioned in section 1.2.

This law can be used to form an equation in terms of as follows:

3. BIBHORR LEEK THEOREM

Theorem 3.0 (Bibhorr leek theorem). For an arc EF (Kālpanik vakra) with radius spanning between the two

mutually perpendicular sides (AE & AF) , the length of Bibhorr leek is given as:

This radius (Bibhorr leek) emanates from the point where AC and AD meet and ends at the point of

intersection of the arcs CD and EF.

FIGURE 3. Diagram for Bibhorr leek theorem.

– A B

C

E D E A

F F

(from Assertions 1.1 & 1.2)

(Equation 2.1)

A B

C

D

(Bibhorr vakra)

E

F

क

(Kālpanik vakra)

J

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Assertion 3.1. When , Bibhorr leek is equal to half the sum of AC and AD. Mathematically, written

as:

Proof.

1. Consider a right triangle ABC such that AB = AC. That is

FIGURE 4. Diagram for right angle triangle when .

2. For such a triangle ∠ ABC = ∠ ACB i.e. = ऊ = 45º and

3. = i.e. = (from Baudhayan/Pythagoraean theorem)

4. Now for this triangle assuming = 90 from Bibhorr law.

5. Solving for , we get =

=

(Putting = ऊ = 45º and = ) [from steps 2 and 3]

6. Simplifying above equation, we get

7. Now, from Assertion.3.1

8. Simplifying Step 7 we get

[From steps 1,3]

9. From steps 6, 7 and 8, it is verified that when then

. Thus Assertion 3.1 is

verified.

Assertion 3.2. When Bibhorr leek is equal to the sum of AC and AD per unit ( 2+ज ).

ज

Derivation. When , then

In order to balance this inequality, a variable ज is added to 2.

ज

Hence, Assertion 3.2 is derived.

Assertion 3.3. The variable ज is given as:

ज =

Derivation. Considering figure 3, the variable ज is directly proportional to the difference between CJ and

half the sum of AD and AC.

A B

C

D E

F J

K

ऊ

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ज ∝

-

[Eq. 3(a)]

The variable ज is inversely proportional to the sum of AD and AC.

ज ∝

[Eq. 3(b)]

From Eq.3(a) and Eq.3(b), we have

ज =

Hence, deriving the assertion

Derivation for expression . From figure 3, following results are evident:

AD = – (Equation 3.1)

AC = (Equation 3.2)

AJ =

(Equation 3.3)

CJ = (Equation 3.4)

From Equations 3.2, 3.3, and 3.4 CJ becomes:

CJ =

(Equation 3.5)

From Equations 3.1 and 3,2, we have

=

(Equation 3.6)

Recalling Assertion 3.3 Derivation statement : The variable ज is directly proportional to the difference between

CJ and half the sum of AD and AC.

CJ -

=

-

Hence, the required expression is obtained.

Derivation 3.1. From the above assertions, the formula for Bibhorr leek theorem can be derived.

Putting the expression of ज from Assertion 3.3 in result of Assertion 3.2, Bibhorr leek is given as:

[From Baudhayan/Pythagoras’ theorem]

BIBHORR FORMULA : SHORTCUT TO TRIGONOMETRY | BY BIBHORR

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=

Simplifying the above equation, we finally obtain:

Hence, Deriving the Bibhorr leek theorem.

4. BIBHORR FORMULA

Formula 4.0 (Bibhorr formula). For a right triangle ABC with shrav , lambu and chhutku , the angle

opposite AC (Bibhorr angle) is given as:

The resultant angle would be in degrees since R.H.S of the equation incorporates 90º.

Derivation 4.1. From Bibhorr law, we have

From Bibhorr leek theorem, we have

Putting the expression for from Bibhorr leek theorem in the R.H.S of Bibhorr law.

º

or

º

A B

C

( in degrees)

( in radians)

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5. CONCLUSION

It can be concluded that Bibhorr formula simplifies the computations and understanding of the concept of

trigonometry without entangling to any trigonometric function. The formula can used to replace other formulas

and equations that involve trigonometric functions. The formula is deduced from Bibhorr law and hence

only different in its symbolic structure.