31151029036484.pdf

48 PLANE TRIGONOMETRY.
The whole difBculty in this operation consists in the multiplication
of each successive sine or cosine by 2 cos 1' = 1.9999999154; but
.this multiplication is much shortened by observing that
so that if we put
2 cos r = 1-9999999154 = 2 - -0000000846
m = -0000000846
we have 2 cos 1' = 2 — wi and therefore
sin 2' = 2 sin 1' — sin 0' — m sin 1'
sin S' = 2 sin 2' — sin V — m sin 2'
sin 4' = 2 sin 3' — sin 2' — 7n sin 3'
&c.
cos 2' = 2 cos 1' — cos 0' — m cos 1'
cos 3' = 2 cos 2' — cos V — m cos 2'
cos 4' = 2 cos 3' — cos 2' — m cos 3'
&c.
which are computed with great facility.
101. It is not necessary, however, to continue this process beyond
30° ; for by (159) and (162) we have
sin (30° + y) = cosy — sin (30° — y)
cos (30° 4- ?/) = cos (30° — y) — sin y
so that the table is continued above 30° by the simple subtraction
of the sines and cosines under 30° previously found. Thus, making
y successively 1', 2', 3', &c.
sin 30° r = cos 1' - sin 29° 59'
sin 30° 2' = cos 2' - sin 29° 58'
sin 30° 3' = cos 3' - sin 29° 57'
&c.
cos 30° r = cos 29° 59' - sin 1'
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cos 30° 2' = cos 29° 58' - sin 2
cos 30° 3' = cos 29° 57' - sin 3'
&c.
This last process requires to be continued only to 45° since the
sines and cosines of the angles above 45° will be respectively the
cosines and sines of their complements below 45°.