Theory:

We have learnt how to determine the trigonometry ratios for the angle \(0^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\) and \(90^{\circ}\).
 
Now let us learn how to calculate the trigonometric ratios of all the other acute angles using the trigonometric tables.
 
\(1^{\circ}\) \(=\) \(60\) minutes. It is denoted by \({60}'\).
 
\({1}'\) \(=\) \(60\) seconds. It is denoted by \({60}''\)
 
The trigonometric tables provide values, correct to four decimal places, for angles ranging from \(0°\) to \(90°\) and spaced at \({60}′\) intervals. A trigonometric table is made up of three parts.
 
A column on the far left with degrees ranging from \(0°\) to \(90°\), followed by ten columns labelled \({0}'\), \({6}'\), \({12}'\), \({18}'\), \({24}'\), \({30}'\), \({36}'\), \({42}'\), \({48}'\), and \({54}'\).
 
Five columns under the head mean difference has values from \(1\), \(2\), \(3\), \(4\) and \(5\).
 
The appropriate adjustment is obtained from the mean difference columns for angles containing other measures of minutes (other than \({0}'\), \({6}'\), \({12}'\), \({18}'\), \({24}'\), \({30}'\), \({36}'\), \({42}'\), \({48}'\), and \({54}'\)).
 
The mean difference is added in the case of sine and tangent but subtracted in the case of cosine.
 
Trigonometric Table:
 
  
1. Find the value of \(\sin 74^{\circ}{39}'\).
Example:
 
Solution:
 
First, rewrite the given sine value as follows:
 
\(\sin 74^{\circ}{39}'\) \(=\) \(\sin 74^{\circ}{36}'\) \(+\) \({3}'\)
 
Find the value of \(\sin 74^{\circ}{36}'\) from the natural sine table by doing the following step.
 
Check for \(74^{\circ}\) in the extreme left column and \({36}'\) in the top row, the decimal value intersecting the corresponding column and row is the required value of \(\sin 74^{\circ}{36}'\).
 
\(\Rightarrow \sin 74^{\circ}{36}'\) \(=\) \(0.9641\)
 
The value corresponding to \(3\) in the mean difference column gives the value of \({3}'\), which is to be added to the ten thousandth place of the above-determined value.
 
\(\Rightarrow {3}'\) \(=\) \(2\).
 
Therefore, the required sine value is given by:
 
\(\sin 74^{\circ}{39}'\) \(=\) \(\sin 74^{\circ}{36}'\) \(+\) \({3}'\)
 
\(=\) \(0.9643\)
 
 
2. Find the value of \(\cos 34^{\circ}{55}'\).
 
Solution:
 
First, rewrite the given cosine value as follows:
 
\(\cos 34^{\circ}{55}'\) \(=\) \(\cos 34^{\circ}{54}'\) \(+\) \({1}'\)
 
Find the value of \(\cos 34^{\circ}{54}'\) from the natural cosine table by doing the following step.
 
Check for \(34^{\circ}\) in the extreme left column and \({54}'\) in the top row, the decimal value intersecting the corresponding column and row is the required value of \(\cos 34^{\circ}{54}'\).
 
\(\Rightarrow \cos 34^{\circ}{54}'\) \(=\) \(0.8202\)
 
The value corresponding to \(1\) in the mean difference column gives the value of \({1}'\), which is to be subtracted from the ten thousandth place of the above-determined value.
 
\(\Rightarrow {1}'\) \(=\) \(2\).
 
Therefore, the required cosine value is given by:
 
\(\cos 34^{\circ}{54}'\) \(=\) \(\cos 34^{\circ}{54}'\) \(+\) \({1}'\) 
 
\(=\) \(0.8200\)
Reference:
State Council of Educational Research and Training (2018). Mathematics. Term - III Volume 2: Chapter 3 Trigonometry(pg.79 - 84). Printed and Published by Tamil Nadu Textbook and Educational Services Corporation.