Plane trigonometry, by S.L. Loney.

PROPORTIONAL PARTS.

163

If we want to find the sine of an angle which contains
an integral number of degrees and minutes we can obtain
it from the tables. If, however, the angle contain seconds
we must use the principle of proportional parts.
Ex. 1. Given sin 29~ 14' = 4883674,
and sin 29~ 15' = 4886212,
find the value of sin 29~ 14' 32".
By subtraction we have
difference in the sine for 1'= -0002538.
32
difference in the sine for 32" = x 0002538= 00013536,.. sin 29~ 14' 32"= 4883674
+ -00013536
= 48850276.
Since we want our answer only to seven places of decimals we omit
the last 6, and, since 76 is nearer to 80 than 70, we write
sin 29~ 14' 32"= -4885028.
N.B. When we omit a figure in the eighth place of decimals we add
1 to the figure in the seventh place, if the omitted figure be 5 or a number
greater than 5.
Ex. 2. Given cos 16~ 27'= 9590672,
and cos 16~ 28' = 9589848,
find cos 16~ 27' 47".
We note that, as was shewn in Art. 55, the cosine decreases as the
angle increases.
Hence for an increase of 1', i.e. 60", in the angle, there is a decrease
of -0000824 in the cosine.
Hence for an increase of 47" in the angle there is a decrease of
7x -0000824 in the cosine.
47.. cos 16~ 27' 47"= -9590672 -7 x 0000824
= 9590672 - -0000645
= 9590672
- 0000645
= 9590027.
11-2