UPSKILL MATH PLUS

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### 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.