Trigonometry with the help of fingers

Trigonometry is a branch of mathematics which basically deals with the relation of all the three sides of a right angle triangle. The first thing that comes out in our mind when we talk about Right angle triangle is – Pythagoras Triangle. The first reference of Pythagoras triangle is found to be in Katyayana Sulbasutra written about 200 BC. In earlier days in India people used to perform Yagna and for that the Rishis used to make alter in different shapes using the triplets now popularly known as Pythagorean Triplets.

As far as its discovery is concerned it was first discovered by the Greeks to aid in the study of astronomy. The Babylonians established the measurement of angles in degrees, minutes and seconds. From Hipparchus to Ptolemy, Aryabhata all contributed in the development of trigonometry. There are six trigonometric ratios namely– Sine, Cos (ine), Tan (gent), Cot (angent), Sec (ant) and Cosec (ant).

Every trigonometric ratio shows a different relation between two of three sides of a right angle triangle.

Let me remind you of the following concepts of trigonometry, which you have learnt at the secondary level. In a right angle triangle ABC with angle C = 90⁰, the various trigonometric ratios are defined as follows: ---

                            

Sin A = Perpendicular / Hypotenuse                          Cosec A = Hypotenuse / Perpendicular

Cos A = Base / Hypotenuse                                        Sec A = Hypotenuse / Base

Tan A = Perpendicular / Base                                     Cot A = Base/ Perpendicular

How to learn the Mnemonics for trigonometric Values?

There are many mnemonics which will help you to learn the trigonometric ratios of an angle in no time. I shall focus here two – one which is sophisticated in look and one which is commonly been used in Indian schools for long.

·         Some People Have Curly Brown Hair Towards Pulled Back --- Here the bold letters shows the relations. S stands for Sine, C stands for Cos and T stands for Tan whereas P stands for Perpendicular, B stands for Base and H stands for Hypotenuse. This single line helps you to understand the relations of sides in different trigonometric ratios.

Sin A = P/H                Cos A = B/ H                          Tan A = P/ B

·         Let’s learn a very funny mnemonic which will help you to find directly the relation between two sides. The mnemonic is -Pandit Badri Prasad Hari Hari Bole. Let me explain it.

Sin	Cos	Tan
Pandit	Badri	Prasad
Hari	Hari	Bole
Cosec	Sec	Cot

Here, Sin A = P/H, Cos A = B/H and Tan A = P /B where P = Perpendicular, B = Base and H = Hypotenuse

How to learn the trigonometric angles from 0 to 90

After learning the relation of sides of a right angle triangle you must learn the value of angles for different ratios in Trigonometry. The trigonometric angle will help you to understand the application of trigonometry in general life. The slope of an angle, the elevation and depression angle all together are based on the value of angles in different ratios.

First let me illustrate a table for you and later explain you how you can construct that table or find out the value with the help of your fingers.

Ratio / Angle	0	30	45	60	90
Sin	0	½	1/√2	√3/2	1
Cos	1	√3/2	1/√2	½	0
Tan	0	1/√3	1	√3	Not defined
Cot	Not defined	√3	1	1/√3	0
Sec	1	2/√3	√2	2	Not defined
Cosec	Not defined	2	√2	√3/2	1

In the first look it seems that it is tough to learn this table but you can learn it on finger tips with little practice and there is absolutely no need to learn such table at all.

Angles on your finger tips

In multiplication we have also seen the magic of our hand and learnt some basic multiplication. Here we will see how helpful our hand is in memorizing the trigonometric table without pen and paper.

·         Mark your fingers with angles from 0 to 90 as shown in the first figure.

·         The number of fingers to the right of designated angle finger will help you to find the value of cosine whereas the number of fingers to the left part of designated finger will help you to find the value of sine.

·         In picture 2, place the number of fingers either right side of the designated angle finger to find the value of cosine or number of fingers to the left side to find the value of sine.

                   

Figure 1

            

Figure 2

Figure 3

Working Method

You will be astonished to know that these fingers will help you to find the value of designated angle for both sine and cosine simultaneously. Suppose you have to find the value of sin 30 and cos 30, you can find both of these values in seconds.

·        Bend the finger marked with 30 as shown here. Though it is not necessary but for the first time user it is recommended as it will make your calculation flawless.                                       

·        Count the number of fingers to the right and left of the bend finger marked with angle 30.             Fingers to the left of bent finger = 1                                                                                  

Finger showing the value of 30 degrree

    Fingers to the right of bent finger = 3

·        Cos 30 = √3/2                                     and                  sin 30 = √1/2 = ½                                           

Isn’t it an easy method to find the value of different angles of sine and cosine simultaneously? Suppose you have to find the value of –

                  Sin 45 = ?                                Cos 45 = ?

Simply bend the finger with 45 marked on it. Count the number of fingers to the left or right of it. Place these value in square root and divide the result by 2.

Fingers to the left of bent finger = 2                                      Fingers to the right of bent finger =2

Hence, Sin45 = √2/ 2 = 1/√2

            Cos 45 = √2/ 2 = 1/√2

We know that tangent of an angle can be found by simply dividing the ratio of sin and cosine.                 Tan A = Sin A / Cos A

            Tan 30 = Sin30/ Cos30 = ½ / √3/2 = 1/√3

Moreover,

            Sec A = 1/ Cos A                    so                     Sec 30 = 2/√3

            Cosec A = 1/ Sin A                 so                     Cosec 30 = 2/1 = 2

I was teaching this technique of extracting trigonometric table for different angles of Sinϴ and Cosϴ  in DAV school in Bihar few month ago when a student stood and asked me how to extract the value of Tanϴ and Cotϴ with the help of finger technique and then I decided to extend the finger technique of extracting angles for different trigonometric ratios in my book. Let’s see the following two images with the angles written over it.

Tan ϴ = Square root (Fingers to the left to bent finger/ finger to the right to the bent finger)

Cot ϴ = Square root (Fingers to the right to the bent finger / finger to the left to the bent finger)

It means that you can work simultaneously to find the value of any Tan and Cot of angles from 0 to 90 degrees. In the figure 3, finger indicating 30 degree has been bent. You can see that there are 1 finger to the left of bent finger and 3 fingers to the right side.

Hence, Tan 30 = 1/ √3                        and                  cot 30 = √3/1 = √3

                         

                                                       Figure 4

In Figure 4, finger with angle 45 marked on it has been bent. As you can see in the picture available here that there are equal numbers of fingers to the both side of bent finger. 

Hence, Tan 45 = square root of (2/2) = 1

            Cot 45 =   square root of (2/2) = 1

  I do hope you can learn the trigonometric table of different angles with the help of above technique in no time. Even in examination hall you can use this technique in case you forget a particular value in seconds. Try to practice it more and more so that you can automatically learn the value of different angles for different trigonometric ratio.

Practice Problem

a)      Sin 60 = ?                                b) Sec 45 = ?                           c) Cot 90 = ?

d) Tan 45 = ?                                 e) Cos 60 = ?                           f) Cosec 45 = ?           

Angles in Different Quadrants

After learning the values of 0 to 90 for different trigonometric ratio, we should learn its extension. The following mnemonic will help you to find the values of any trigonometric ratios from 0 to 360 in no time.

Add Sugar To Coffee :- In Trigonometry, the student of class 11 feels uneasy when they are asked to remember the Trigonometric Table above than 90 degree but this one liner mnemonic Add Sugar To Coffee  will certainly help you to find the value of trigonometric value of any degree. Let’s see its working.

The above representation will help you to find the value of angles from 0 degree to 360 degrees and more. The only point to remember is that---   

 a) For 90 ± ө , 270 ± ө, trigonometric function changes into sinө ↔cosө, tanө ↔ cotө and             secө ↔ cosecө         

b) For 180 ± ө, 360 ± ө, there will be no change in the result. Let us understand it more vividly.

In 1^st Quadrant

In the first quadrant the angle should be less than 90 degree and as discussed above for values less than 90 the trigonometric function changes to its complementary trigonometric function. Hence

Sin(90 – ө ) = cosө

Cos (90 – ө) = sinө

Tan (90−ө) = cotө and so on.

For Second Quadrant

 In second quadrant the value should be either 90 + ө or 180−ө and the value of sine remain positive and so will be with its reciprocal cosec and for the rest trigonometric ratio the result will bear a negative sign. I have discussed above that for 180±ө and 360±ө the trigonometric ratio will remain unchanged so let’s see how does the value of trigonometric ratio changes for certain degrees that lies in 2^nd quadrant.

 Sin (90+ө) = cosө                                          sin (180−ө) = sinө      

Cos(90 + ө) = − sinө                                       cos (180 –ө) = − sinө

Tan (90 + ө) = − cotө                                     tan (180 −ө) = −tanө

 I do hope the reader will enjoy the method and take self-initiative to find the value of trigonometric function in 3^rd and 4^th quadrant.

The above discussion can be easily understood in the following diagrams.

 Illustration

Sin 30 = ½ = sin 150 (Sum of angles is 180, hence no sign change in result)

Sin 150 = ½ but sin 210 = - ½ (Sum of angles is 360, hence negative sign in result)

Sin 60 = √3/2 = sin 120 (Sum of angles is 180, hence no sign change in result)

Sin 120 = √3/2 but sin 240 = -√3/2 (Sum of angles is 360, hence negative sign in result)

Let’s now move the discussion to find the value of cosine.

The above discussion can be easily understood in the following diagrams.

Illustration

Cos 60 = ½                        Cos 120 = - ½ (Sum of angle = 180 so sign changes in Cosine)

Cos 45 = 1/√2                   Cos 135 = - 1/√2 (Sum of angle = 180, so sign changes in Cosine)

Cos 120 = - ½                   Cos 240 = ½ (Sum of angle = 360, so sign changes in Cosine)

Cos 135 = - 1/√2               Cos 225 = 1 / √2 (Sum of angle = 360, so sign changes in Cosine)

For more information Read Mathematics Made Easy by Rajesh Kumar Thakur published by Rupa Publication. Books available on Amazon , Flipkart and other websites

Rajesh Kumar Thakur

rkthakur1974@gmail.com