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11. Find the number whose logarithm is 4.7989672.

Ans. 62945.876.

70. Use of Trigonometric Tables. — Trigonometric Tables are of two kinds, -Tables of Natural Trigonometric Functions and Tables of Logarithmic Trigonometric Functions. As the greater part of the computations of Trigonometry is carried on by logarithms, the latter tables are by far the most useful.

We have explained in Art. 27 how to find the actual numerical values of certain trigonometric functions, exactly or approximately.

1 Thus, sin 30° = } that is, .5 exactly. Also, tan 60°=V3; that is, 1.73205 approximately.

A table of natural trigonometric functions gives their approximate numerical values for angles at regular intervals in the first quadrant. In some tables the angles succeed each other at intervals of 1", in others, at intervals of 10", but in ordinary tables at intervals of 1': and the values of the functions are given correct to five, six, and seven places. The functions of intermediate angles can be found by the principle of proportional parts as applied in the table of logarithms of numbers (Arts. 68 and 69).

It is sufficient to have tables which give the functions of angles only in the first quadrant, since the functions of all angles of whatever size can be reduced to functions of angles less than 90° (Art. 35).

71. Use of Tables of Natural Trigonometric Functions. — These tables, which consist of the actual numerical values of the trigonometric functions, are commonly called tables of natural sines, cosines, etc., so as to distinguish them from the tables of the logarithms of the sines, cosines, etc.

We shall now explain, first, how to determine the value of a function that lies between the functions of two consecutive angles given in the tables; and secondly, how to determine the angle to which a given ratio corresponds.

72. To find the Sine of a Given Angle.
Find the sine of 25° 14' 20", having given from the table

sin 25° 15' =.4265687
sin 25° 14' =.4263056

diff. for 1' = .0002631

Let d= diff. for 20"; and assuming that an increase in the angle is proportional to an increase in the sine, we have

60 : 20::.0002631: d.
20 x .0002631

.0000877.

60
... sin 25° 14' 20" = .4263056 +.0000877

=.4263933.

.. d

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NOTE. - We assumed here that an increase in the angle is proportional to the increase in the corresponding sine, which is sufficiently exact for practical purposes, with certain exceptions.

73. To find the Cosine of a Given Angle.
Find the cosine of 44° 35' 25", having given from the table

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cos 44° 35' = .7122303
cos 44° 36' = .7120260

diff. for l'=.0002043

observing that the cosine decreases as the angle increases from 0° to 90°.

Let d= decrease of cosine for 25"; then

60 : 25 ::.0002043: d.

1

..

25
d = X.0002043 = .0000851.

60

... cos 44° 35' 25'' = .7122303 – .0000851

= 7121452.

Similarly, we may find the values of the other trigonometric functions, remembering that, in the first quadrant, the tangent and secant increase and the cotangent and cosecant decrease, as the angle increases.

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cos 74° 45'= .4265687,
cos 74° 46'=.4263056;
cos 74° 45' 40".

find

.4263933.

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74. To find the Angle whose Sine is Given.

Find the angle whose sine is .5082784, having given from the table

sin 30° 33' = .5082901
sin 30° 32' = .5080396

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Let d=diff. between 30° 32' and required angle; then .0002505 :.0002388 :: 60 : d.

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75. To find the Angle whose Cosine is Given.

Find the angle whose cosine is .4043281, having given from the table

cos 66° 9'=.4043436
cos 66° 10' = .4040775

=

diff. for 1'=.0002661

cos 66° 9' = .4043436 given cosine -.4043281

diff. = .0000155

Let d=diff. between 66° 9' and required angle; then .0002661:.0000155 :: 60: d.

:

.. d=

155 x 60

= 3.5.

2661 Required angle is greater than 66° 9' because its cosine is less than cos 66° 9'.

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1. Given

sin 44° 12' =.6971651,

sin 44° 11' = .6969565; find the angle whose sine is .6970886. Ans. 44° 11' 38'.

48° 46' 34".

2. Giver

sin 48° 47' = .7522233,

sin 48° 46' = .7520316; find the angle whose sine is .752140. 3. Given sin 24° 11' = .4096577,

sin 24° 12' = .4099230; find the angle whose sine is .4097559.

=

24° 11' 22".2.

4. Given

eos 32° 31'=.8432351,

cos 32° 32' =.8430787; find the angle whose cosine is .8432.

=

32° 31' 13".5.

5. Given cos 44° 11'=.7171134,

cos 44° 12' = .7169106; find the angle whose cosine is .7169848.

=

44° 11' 38".

=

6. Given cos 70° 32' = .3332584,

cos 70° 31'

.3335326; find the angle whose cosine is .3333333.

70° 31' 43".6.

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