§ 3. COMPUTATION OF FUNCTIONS.

PROB. 1. TO COMPUTE A TABLE OF NATURAL SINES AND COSINES.

(a) For angles 0°... 30°: replace 0 by 1', 2', 3', ... in the formulce of th. 3.

NOTE 1. The fraction : 10,800 once raised to the required powers, first, second, third, ..., and divided by the factorials 1, 2, 3,..., thereafter only simple multiples of the quotients are used. At first but two terms of the series are needed; but later, when is larger, and the series therefore converges less rapidly, more terms must be taken.

E.g., for 30°, 0.52360 nearly;

i.e., by the use of three terms of the series, the sine is found correct to four decimal places, the same degree of accuracy as that assumed for the value of π.

(b) For angles 30°... 45°: replace O' by 1', 2', 3',... in formulæ : sin (30°+0')= cos 0'-sin (30°-0'), [ad. th., sin 30°

cos (30°+0')= cos (30° — 0') — sin O'.

E.g., sin 30° 1' = cos 1' sin 29° 59'

= .999 999 .. .499 75.500 25,

...

sin 30° 2' = cos 2'- sin 29° 58'

= 999 999... -.499 50.500 50,

cos 30° 1' = cos 29° 59' - sin 1'

= .86617.000 29.865 88,

cos 30° 2' = cos 29° 58' - sin 2' = .865 73.

(c) For angles 45°... 90°: apply formula: sin (45° +0')= cos (45° — 0') ;

cos (45° + 0') = sin (45° — 0').

E.g., sin 45° 1' = cos 44° 59′ = .70731, sin 45° 2' = cos 44° 58' = .70752,

cos 45° 1' = sin 44° 59' = .706 90,

cos 45° 2' sin 44° 58′ = .706 70.

=

[II th. 8

NOTE 2. VERIFICATION. The results are tested in many ways:

.. from cos 45°, =√1⁄2, are found in succession the sines and cosines of 22° 30', 11° 15',....

from cos 30°, = √√√3, are found in succession the sines and cosines of 15°, 7° 30', ..

So

....

[II th. 14

[II th. 16 cr.

[II th. 8

sin 36° = cos 54°,

... 2 sin 18° cos 18° 4 cos3 18° - 3 cos 18°,

... sin 18° (5-1), cos 18° = √(10+2 √5);

thence in succession the sines and cosines of 9°, 4° 30′, 2° 15',

(c) From cos 36° cos2 18° - sin2 18°=‡ (√5+1),

=

and sin 36° = √(1 − cos2 36°) = √ (10 − 2 √5),

are found the sine and cosine of (36° — 30°), i.e., of 6°, thence in succession the sine and cosine of 3°, 1° 30', 45',

....

(d) From sin (36°+0') — sin (36° 0′), 2 cos 36° sin ' =(√5+1) sin 0', subtract sin (72°0') sin (72° — 6'), 2 cos 72° sin '

then,

a formula (Euler's) that serves to test the sines of all angles from 0° to 90°, if to 0' be given the different values from 0° to 18°.

In the same way may be used either of the test formulæ found in II § 8, exs. 9-12.

PROB. 2. TO COMPUTE TABLES OF NATURAL TANGENTS, COTANGENTS, SECANTS, AND COSECANTS.

Divide the sines of the angles, in order, by the cosines, each by each; the cosines by the sines; unity by the cosines; unity by the sines:

or, replace by 1', 2', 3', ... in the formula of th. 3, cr. 1.

PROB. 3. TO COMPUTE TABLES OF LOGARITHMIC FUNCTIONS.

From a table of logarithms of numbers take out the logarithms of the natural sines and cosines :

or, replace 0 by 1', 2', 3', ... in the formulee of th. 3, cr. 2.

Subtract the logarithmic cosines from the logarithmic sines; the logarithmic sines from the logarithmic cosines; the logarithmic cosines and sines from 0.

NOTE. 1. A more rapid method, applicable also to making tables of natural functions, and many others, is this:

Take out the functions of three, four, or more angles at regular intervals, and find their several "orders of differences"; then, by the algebraic "method of differences," find the successive terms of

the series of logarithms, and interpolate for other angles lying bctween those of the series. Verify at intervals by direct computation. [o. w. J. alg. XII §§ 10, 11

For safety, four-place tables must be computed to six places; five-place tables to seven places, and so on.

When the terms of any order of differences are constant, or differ very little, the rule that follows may be applied to form new terms of the series:

Add the constant difference to the last difference of the next lower order, that sum to the last difference of the next lower order, and so on till a term of the series is reached.

In the example that follows, the numbers below the heavy rules are got by successive addition :

2. By interpolation in the table above find the logarithmic

sines of 18° 1', 18° 2', 18° 3', 18° 4', ....

3. From the logarithmic sines of 8°, 8° 10', 8° 20', 8° 30',

taken from the table, find the several orders of differ

ences, thence find three more terms of the series.

4. Interpolate to minutes between 8° 10' and 8° 20'.

§ 4. RELATIONS BETWEEN PLANE AND SPHERICAL

TRIANGLES.

After the definition of the trigonometric functions and the statement of their relations, all the properties of the right spherical triangle, and of the plane triangle (oblique and right), may be derived from those of the oblique spherical triangle. Such a development of the subject presents the principles of trigonometry in their most general form, and teaches the student to take these general propositions and, by successive steps, to draw out and state in their logical order the special propositions that are included in them. This mutual relation of the general and the particular not only helps the intellect to grasp these propositions, but also helps the memory to retain them.

The order of development is this:

To state and prove the general properties of the oblique spherical triangle, counting it the most general form of the triangle.

To derive the properties of the right spherical triangle, counting it a special case of the oblique spherical triangle, wherein one angle is a right angle.

To derive the general properties of the oblique plane triangle, counting it a special case of the spherical triangle, wherein the radius of the sphere has become infinite and the arcs straight lines.

To derive the properties of the right plane triangle, counting it a special case of the oblique plane triangle, wherein one of the angles is a right angle; or of the right spherical triangle wherein the arcs are straight lines.