Finding tan Value from Trigonometric Table

We know the values of the trigonometric ratios of some standard angles, viz, 0°, 30°, 45°, 60° and 90°. While applying the concept of trigonometric ratios in solving the problems of heights and distances, we may also require to use the values of trigonometric ratios of nonstandard angles, for example, sin 62°, sin 47° 45′, cos 83°, cos 41° 44′ and tan 39°. The approximate values, correct up to 4 decimal places, of natural sines, natural cosines and natural tangents of all angles lying between 0° and 90°, are available in trigonometric tables.

Trigonometric tables consist of three parts.

(i) On the extreme left, there is a column containing 0 to 90 (in degrees).

(ii) The degree column is followed by ten columns with the headings

0′, 6′, 12′, 18′, 24′, 30′, 36′, 42′, 48′ and 54′ or

0.0°, 0.1°, 0.2°, 0.3°, 0.4°, 0.5°, 0.6°, 0.7°, 0.8° and 0.9°

(iii) After that, on the right, there are five columns known as mean difference columns with the headings 1′, 2′, 3′, 4′ and 5′.

Note: 60′ = 60 minutes = 1°.

1. Reading the values of tan 38°

To locate the value of tan 38°, look at the extreme left column. Start from the top and move downwards till you reach 38.

We want the value of tan 38°, i.e., tan 38° 0′. Now, move to the right in the row of 38 and reach the column of 0′.

We find 0.7813.

Therefore, tan 38° = 0.7813.

2. Reading the values of tan 38° 48′

To locate the value of tan 38° 48′, look at the extreme left column. Start from the top and move downwards till you reach 38.

Now, move to the right in the row of 38 and reach the column of 48′.

We find 8040 i.e., 0.8040

Therefore, tan 38° 48′ = 0.8040.

3. Reading the values of tan 38° 10′

To locate the value of tan 38° 10′, look at the extreme left column. Start from the top and move downwards till you reach 38.

Now, move to the right in the row of 38 and reach the column of 6′.

We find 7841 i.e., 0.7841

So, tan 38° 10′ = 0.7841 + mean difference for 4′

= 0.7841

+        19  [Addition, because tan 38° 10′ > tan 38° 6′]

0.7860

Therefore, tan 38° 10′ = 0.7860.

Conversely, if tan θ = 0.9228 then θ = tan 42° 42′ because in the table, the value 0.9228 corresponds to the column of 42′ in the row of 42, i.e., 42°.

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