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“”Like the crest of the peacock, like the gem on the head of

a snake, so is mathematics at the head of all knowledge.

Vol 1: Trigon ArithmeticThis course will help you in solving traditional Trigonometric, Cartesian and related problems without using the complex formulae that a student has to memorize. This method uses very simple yet seemingly magical ways to solve the same complex problems.

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INTRODUCTION TO TRIGON(S) • In this course, the term ‘trigon’ refers to a set of 3

numbers which satisfy the following condition:• The sum of squares of the first two numbers is equal to the

square of the third number• e.g. 3,4,5 form a trigon since 32 + 42 = 52

• Hence all trigons satisfy the Pythagoras theorem:

• Thus, a set of any 3 numbers satisfying this property is a trigon

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TRIGONS AND PYTHAGORAS THEOREM• The theorem of Pythagoras was known long before Pythagoras

(who lived c540 B.C.), the earliest statement of it being in the Indian Sulba-sutras dated c800 B.C.• For example, to quote from Katyanana Sulba-sutra , “the square of

the diagonal of an oblong is equal to the square of both it sides”.• In fact, the Pythagoras theorem can be obtained from the Vedic

Formula “ “ which means the sum of products is the product of the sum. Here products means square and the hypotenuse is the sum of squares of the other two sides.• The dimensions of altars in ancient India were based on perfect

trigons.

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TRIGON BASICS• A trigon in which all 3 elements are rational numbers is called

a rational trigon• e.g. 3,4,5; 8,15,17; 1.5,2,2.5; -4,3,5 etc.

• 2,3,13 is not a rational trigon since 13 is an irrational number.

• Any trigons that are multiples or fractions of other trigons are called equal trigons. • e.g. 4,3,5; 8,6,10; 2, 1.5, 2.5; 12,9,15 are equal trigons

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TRIGON BASICS (CONTD.)• A rational trigon is called a primary trigon if all 3 elements

are whole numbers and have no common factor other than 1.

• In every family of equal perfect trigons, there is one and only one primary trigon.

• For example, among 4,3,5; 8,6,10; 2, 1.5, 2.5; and 12,9,15 4.3,5 represent the primary trigon.

• 3,4,5 is the complimentary trigon for 4,3,5. Hence to obtain a complementary trigon, we just transpose the first two elements.

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TRIGON NOTATION• Since trigons satisfy Pythagoras theorem, we can

represent right-angled triangles using trigons and vice-versa.

• The notation ) 4,3,5 represents the angle (between sides of length 4 & 5) of the right-angled triangle with sides 4,3,5 where 4 is the base, 3 is the perpendicular (height) and 5 is the hypotenuse.

• Thus, Angle) Base, Perpendicular, Hypotenuse is the general notation for a trigon.

𝜃3

4 5

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BENEFITS OF TRIGON NOTATION• With this notation – ) 4,3,5 all the trigonometric ratios are

immediately available.

• For example:• = B/H =4/5• = P/H = 3/5• = P/B = 3/4

𝜃3

4 5

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TRIGON ARITHMETIC

• The Trigon Arithmetic mainly includes:• Trigon Addition• Trigon Subtraction

•Other Trigon Arithmetical operations frequently used are:• Double Angle• Half Angle

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TRIGON ADDITION•Consider the following two trigons:•

• where B= Base, P=Perpendicular/Height and H=Hypotenuse

• Then the trigon addition is calculated as- B P H b p h + Bb-Pp Pb+Bp Hh

•Hence, when we add the trigons for the angles and , we get the trigons for the angle

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TRIGON ADDITION – VERTICALLY & CROSSWISE• The Trigon Addition is actually an application of “Vertically &

Crosswise” formula of Vedic Maths. B P H B P H B P H b p h b p h b p h Bb-Pp Pb+Bp Hh Base Perpendicular Hypotenuse

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TRIGON ADDITION – NUMERICAL•Consider the following two trigons:

• Then the trigon + is calculated as- ), ), ()

•Hence the trigon for the angle +is 16, 63, 65

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TRIGON ADDITION – GEOMETRICAL ILLUSTRATION•Consider the following two trigons:

• Geometrical illustration is on the right• The addition is done as follows-

435

1517

8

θ𝜙

36

77

85

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APPLICATIONS OF TRIGON ADDITION•Given =3/5 and =7/24 find-

In modern mathematics, to solve these kinds of equations we have to memorize lots of formulae. However, it is not needed when we apply the Vedic formulae “Sum of Products = Product of the Sum” [SoP=PoS], and “Vertically & Crosswise” together in trigon form. 36

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APPLICATIONS OF TRIGON ADDITION (CONTD.)

•Clearly, trigon and trigonometric ratio are related. Here we have been given =3/5, i.e. P=3 and H=5 and =7/24, i.e. p=7 and b=24. •Using SoP=PoS and V&C, we get

The primary trigon for 75,100,125 is 3,4,5. Hence we have ) 3,4,5

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APPLICATIONS OF TRIGON ADDITION (CONTD.)

•Now that we have ) 3,4,5, the following can be very easily calculated.• = P/H = 4/5• = H/B = 5/3• = P/B = 4/3• = B/P = 3/4

Hence, given any two or more trigonometric ratio, we can very easily obtain any trigonometric ration of the form .

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TRIGON ADDITION – COMPLEMENTARY TRIGONS•Given =3/5 and =5/4 find • Using SoP=PoS, we have ) 4,3,5 and ) 3,4,5

• Since we need to calculate only we don’t need to calculate the base/1st element of the trigon. Similarly, if we needed to find , we wouldn’t need to evaluate the 3rd element of the trigon.• Here we clearly have = 25/25 = 1.

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TRIGON ADDITION – COMPLEMENTARY TRIGONS• In the last example, if we calculate the first element we get-

• The primary trigon for 0,25,25 is 0,1,1 which implies that the base of the triangle is Zero. This is because 0,1,1 represent the angle 90º (or /2c)

• Any two angles - and are called complementary angles if 36

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TRIGON ADDITION – COMPLEMENTARY TRIGONS•Here we have

• The trigon for is 4,3,5 and trigon for is 3,4,5.

•Here 4,3,5 is complementary trigon of 3,4,5.

• In terms of trigon addition if sum of two trigon is 0,1,1 they are complementary trigons.

• 3,4,5 is the complimentary trigon for 4,3,5. Hence to obtain a complementary trigon, we just transpose the first two elements. 36

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TRIGON ADDITION – MORE EXAMPLES

36

B A

D

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TRIGON ADDITION – DOUBLE ANGLE• Consider the angle ) B,P,H then the double angle can be

obtained as-

• For example,

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DOUBLE ANGLE – MORE EXAMPLESUsing the formula we can compute trigons for double angles very easily-

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TRIGONS AND COORDINATE GEOMETRYConsider the double angle for the trigon 3,4,5 -

Here we see that the first element, i.e., the base is negative. This is because the resultant angle is obtuse.

-7

24

Y

X

2∅ ∅∅

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TRIGONS AND COORDINATE GEOMETRYSimilarly, the same trigon values can be in any of the four quadrants each representing an angle in that quadrant-

-7,24,25 Y

X

7,24,25

7,-24,25

-7,-24,25

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TRIGON SUBTRACTION•Consider the following two trigons:•

• where B= Base, P=Perpendicular/Height and H=Hypotenuse

• Then the trigon subtraction is calculated as- B P H b p h Bb+Pp Pb-Bp Hh

•Hence, when we subtract the trigon for the angle from the angle , we get the trigons for the angle

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TRIGON SUBTRACTION – V&C

• The Trigon Subtraction is another application of “Vertically & Crosswise (V&C)” formula of Vedic Maths.

B P H B P H B P H b p h b p h b p h Bb+Pp Pb-Bp Hh Base Perpendicular Hypotenuse

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TRIGON SUBTRACTION – NUMERICAL•Consider the following two trigons:

• Then the trigon is calculated as- ), ), ()

•Hence the trigon for the angle is 56, -33, 65

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TRIGON SUBTRACTION – GEOMETRICAL ILLUSTRATION•Consider the following two trigons:

• Geometrical illustration is on the right• The addition is done as follows-

4

515

17

8

36

85

133

84

𝜙-θ

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APPLICATIONS OF TRIGON SUBTRACTION•Given =7/25 and =4/5 where is acute and is obtuse, find

• It is to be noted that for obtuse angles the base is negative.

Hence = 75/125 = 3/5

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APPLICATIONS OF TRIGON SUBTRACTION (CONTD.)

•Given =24/7 and is acute find

Hence =

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APPLICATIONS OF TRIGON SUBTRACTION (CONTD.)

•Given and ,find

Hence =

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TRIGON ARITHMETIC – HALF ANGLE• Consider the angle ) B,P,H then the half angle can be

obtained as-

Where, the 3rd element, i.e., the hypotenuse can be obtained by the first two elements. The sign of the half angle is decided based on the quadrant of the angle.36

OR

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HALF ANGLE –EXAMPLEUsing the formula we can compute trigons for half angles very easily.

Given that )-3-4,5 the half angle can be found as given below:

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OR

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HALF ANGLE –EXAMPLEIf find .Solution:

Here we have =7/25 and Hence = 7/75

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APPLICATIONS OF TRIGON:• This presentation has only covered the mere basics of trigon

using Vedic Mathematics formulae V&C and SoP=PoS.• This basic arithmetic can be used to solve all kinds of

Trigonometric equations, Inverse and Hyperbolic Trigonometric functions and Solution to Triangles (e.g. Sine Rule) and much more.• Solutions to many of the questions asked in JEE, JEE Advanced,

Quizzes, GATE etc. can be done very quickly using these methods.• The same thing can be used in Coordinate Geometry and various

other branches of mathematics. 36

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TRIGON APLLICATIONS IN COORDINATE GEOMETRY• Rotate point P(4,3) anti-clockwise through an angle of 90o about the

origin.

• Using Trigon Addition, we have -

So the position of the rotated point Q is (-3,4).• It is to be noted that if the rotation is anti-clockwise we add the triple and

if it is clockwise we subtract the triple.

• Starting with the simplest of problems, even some of the most complex ones can be easily solved by applying the formulae of Vedic Mathematics.36

Q