By Dr Bhajan Singh Lark

Retd Prof (Chemistry) Guru Nanak Dev University, Amritsar, Punjab, India.

 While solving problems in science we many a times have to use t– ratios say like sin of certain angle. Naturally we look up for tables of t- ratios. For example while solving problems on X- rays we have to use sin of certain angle in Bragg’s (1) equation

                                         nʎ = 2d sin θ.                                                         … (1)

But the table is not easily available. Here I am suggesting a method to find t- ratios of any angle quite precisely (and that too) without using a table.

This method involves mainly the following trigonometric corollary that:

Sin θ = θ when θ is small (θ is measured in radians)

How small should be the angle for this corollary to hold good.  It will depend upon as to how accurately we need the value of sin θ. Generally we use the value of sin θ correct up to 4 places of decimal which are given usually in mathematical tables(2). Here the procedure to determine the value of sin θ of any angle between 0-90^o is explained. 

We know that π radians equal to 180^o. Thus the value of a radian in degrees will depend upon the value of π. For accurate value of sin of an angle up to 4 places of decimal the value of π = 3.1416 1.e., taken up to 4 places of decimal works very well.

Understandably if 3.1416 (π) radians  =  180^o

Then one radian     = 180^o/3.1416 = 57.296^o.

Let us check if 1/10th of a radian which is equal to 5.7296^o rounded to = 5.730 is small or not for the above relation 1 to hold.

Sin of 0.1 radian should be equal to 0.1.  As given by scientific calculator, the value of sin 5.730 is 0.0998 which is the same as given in 4 fig. sin Tables. Thus the present method gives the value of sin 5.73, 0.0002 more than the table value. It is clear form table I that the sin of any angle up to 4^o agrees very well with that predicted by the relation 1.Further on this relation predicts a value a little higher than the table or the calculator i.e. real value of sin θ and as the angle becomes higher and higher, the deviation becomes bigger and bigger.

Table 1.  Comparison of predicted values of sin θ with the calculator values

Theta (Degrees) θ 1 2 3 4 5 5.73 6 7 8 9 10

sin θ = θ  (radians) _________________ 0.01745 0.0349 0.0524 0.0698 0.0873 0.1000 0.1047 0.1222 0.1396 0.1571 0.1745

Sin θ (Calculator value) ___________________ 0.01745 0.03490 0.05234 0.06976 0.08716 0.09984 0.10453 0.12187 0.13917 0.15643 0.17365

Difference ________________ 0.00000 0.00000 -0.00006 +0.00004 +0.00014 +0.00016 +0.00017 +0.00033 +0.00043 +0.00067 +0.00085

Below I give a method to know the value of sin θ of any angle having θ greater than 4^o 

accurate up to 4 or more places after the decimal.  The method utilizes the following identities.

sin2θ = 2 sinθ cos θ  = 2sin θ (1-sin² θ)^0.5                               (2)

And when θ  is small, i.e. sin θ  = θ   therefore relation 2.

becomes,                   sin2θ = 2 θ (1- θ ²)^0.5                                                (3)

sin 3θ = 3sinθ – 4sin³θ =  3θ – 4θ³                                                  (4)

sin(A+B) = sinAcosB+ cosAsinB

=  A(1-B² )^0.5 + B(1-A² )^0.5                                                  (5)

sin(A-B) = sinAcosB- cosAsinB

=  sinA(1-B² )^0.5 + cosA (1-A² )^0.5                                   (6)

 The method involves breaking up the given angle so as to have angles smaller than preferably 3 or 4 degrees so that the sin value is known accurately up to 5 places of decimals and then by the help of any of the identity or of a judicious combination of these gives the sin of the desired angle . Let us find the value of sin 5. It can be arrived as follows,

                              sin 5 = 3sin (5/3) – 4sin³ (5/3)

Angle 5/3 when expressed in radians = 0.01745x 5/3

Thus 3 sin 5/3 degrees = 3x 0.01745x 5/3 =  5×0.01745= 0.08725

Similarly 4sin³ (5/3)   = 4x(0.01745)³  =  0.00002

                          And thus  sin 5 =  0.08725     –  0.00002

= 0.08723 = 0.0872 (up to 4 places of decimals) and this value agrees quite well with the table value of 0.0872 and the calculator value of .08716

Below we give calculated values for sin of certain angles and compare them with the table values and refer to the identity used.

Table 2

Sin angle	Identity	θ	Calculated Sin angle	Table value
Sin 4	3	2x 0.017453	0.0698	0.0698
Sin 4	4	4/3(0.01745)	0.0698	0.0698
Sin 5	3	5/2(0.01745)	0.0872	0.0872
Sin 6	4	2x 0.017453	0.1045	0.1045
Sin 7	5	4.3(0.01745)	0.1219	0.1219
Sin 8	4	8/3(0.01745)	0.1392	0.1392
Sin 8.4	4	8.4/3( 0.01745)	0.1461	0.1461
Sin 9	4	9/3(0.01745)	0.1564	0.1564
Sin 10	4	10/3(0.01745)	0.1737	0.1736

 For values of sin θ of angles higher than 10 degrees we can use the values of angles up to 10 degrees as given in table 2 and then use  any of the relations 3-5 given above.

For angles – 30,45,60,90 the values of sin θ are straight forward and for any angle near these angles, any of the standard relations

        sin(A+B) = sinA cosB + cos A sin B

 may be used.     Here (angle) A = 30, 45,60  i.e. the angles for which the  t values  are known and B is a small angle less than 4 for which sin θ = θ. When θ is in radians.

sin θ values for some selected angles are compared with the table values  in Table 3.

Table 3.

Sin θ	Break up	Calculated value	Table value
Sin 32	30 + 2	0.5299	0.5299
Sin 42	45 – 3	0.6691	0.6691

Thus we see that we can calculate quite precisely sin θ of any angle and certainly then we can calculate from that value, the value of any of the other t ratio/s of the given angle by usual trigonometric relations.

References:

1.States of Matter by B.S.lark and S. Joseph, Vishal Pub. Co. Jallandhar . 2011.

2.The Spectrum of Mathematics, by K.K. Gupta and others, Sharma Publications VII Edition,1997.