Page 24 - DMTH201_Basic Mathematics-1

Unit 2: Trigonometric Functions-II

2.1 Computing Trigonometric Functions Notes
Ptolemy (100–178) produced one of the earliest tables for trigonometry in his work, the Almagest,
and he incorporated the mathematics needed to develop that table. It was a table of chords for
every arc from 1/2 through 180 in intervals of 1/2 . Also he explained how to exclaim between
the given angles. let’s look at how to create tables for sines and cosines using his methods. First,
based on the Pythagorean theorem and similar triangles, the sines and cosines of certain angles

can be computed directly. In particular, you can directly find the sines and cosines for the angles
30 , 45 , and 60 as described in the section on cosines. Ptolemy knew two other angles that could
be constructed, namely 36 and 72 . These angles were constructed by Euclid in Proposition IV.10
of his Elements. Like Ptolemy, we can use that construction to compute the trig functions for
those angles. At this point we could compute the trig functions for the angles 30, 36, 45, 60, and
72 degrees, and, of course we know the values for 0 and 90 degrees, too.
Keeping in mind the sine of an angle, the cosine of the complementary angle
cos t = sin (90 – t) sin t = cos (90 – t)
So you have the trig functions for 18 and 54 degrees, too.
Use of the half–angle formulas for sines and cosines to compute the values for half of an angle if
you know the values for the angle. If it is an angle between 0 and 90 , then
sin t/2 = ((1 – cos t) / 2) cos t/2 = ((1 + cos t) / 2)

Using these, from the values for 18, 30, and 54 degrees, you can find the values for 27, 15, and 9
degrees, and, therefore, their complements 63, 75, and 81 degrees.
With the help of the sum and difference formulas,
sin (s + t) = sin s cos t + cos s sin t
sin (s – t) = sin s cos t – cos s sin t

cos (s – t) = cos s cos t + sin s sin t
cos (s + t) = cos s cos t – sin s sin t

you can find the sine and cosine for 3 (from 30 and 27 ) and then fill in the tables for sine and

cosine for angles from 0 though 90 in increments of 3 .
Again, using half–angle formulas, you could produce a table with increments of 1.5 (that is, 1
30’), then 0.75 (which is 45’), or even of 0.375 (which is 22’ 30”).

2.1.1 Addition and Subtraction of Trigonometric Functions

Earlier we have learnt about circular measure of angles, trigonometric functions, values of
trigonometric functions of specific numbers and of allied numbers.

You may now be interested to know whether with the given values of trigonometric functions of
any two numbers A and B, it is possible to find trigonometric functions of sums or differences.

You will see how trigonometric functions of sum or difference of numbers are connected with
those of individual numbers. This will help you, for instance, to find the value of trigonometric

functions of /12 can be expressed as /4 /6
5 /12 can be expressed as /4 + /6
How can we express 7 /12 in the form of addition or subtraction?
In this section we propose to study such type of trigonometric functions.

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