The tangent of the sum of two angles is found thus:

=

sin. (A+B) __sin. A cos. B+cos. A sin. B cos. (A+B)

cos. A cos. B - sin. A sin. B

Dividing both terms of the fraction by cos. A cos. B,

846.-1. Demonstrate formula II in the same manner as formula 1, and both of them for those cases where the angles are not acute. Observe in what quarters the sine and cosine are negative.

2. Express each formula in ordinary language; for example: the sine of the sum of two angles is equal to the sum of the products of the sine of each by the cosine of the other.

3. Demonstrate cos. 12°

(√√30 + 6 √5+ √5 −1.)

FUNCTIONS OF MULTIPLES AND PARTS OF ANGLES.

847. In the formulas of the sine, the cosine, and the tangent of the sum of two angles, suppose BA; then,

By substituting (n-1)A for B in the original formulas, sin. nA, cos. nA, and tan. nA may be expressed in functions of A and of (n-1)A. Thus, when the functions of

A are known, the functions of 2A, 3A, etc., may be calculated.

2

Since cos. A sin,' A=1 (838), we have

2

cos. 2A = 1-2 sin. A; also, cos. 2A=2 cos.2 A-1.

These formulas being true for all angles, A may be substituted for A. Then, transposing,

2 sin.2 A1 cos. A, and 2 cos.? ¿A = 1+ cos. A.

By these formulas, from the cosine of an angle, may be calculated the sine and cosine of its half, fourth, eighth, etc.

2. What is the value of sin. 15°; cos. 3°; sin. 1° 30'?

FORMULAS FOR LOGARITHMIC USE.

849. In order to render a formula fit for logarithmic calculation, products and quotients must be substituted for sums and differences. This may frequently be done by means of the formulas which follow.

The formulas for the sine and cosine of (A+B) become, by adding the third to the first, subtracting the third from the first, adding the second to the fourth, and subtracting the second from the fourth (845),

sin. (A+B)+ sin. (A-B) = 2 sin. A cos. B,

sin. (A+B) sin. (AB)=2 cos. A sin. B,

-

cos. (A+B)+cos. (A — B) = 2 coş. A cos. B,

(1.)

(II.)

(III.)

cos. (AB) - -cos. (A+B)= 2 sin. A sin. B, (IV.)

=

In the above, let A+B C, and A — B=D; whence, A = (C+D), and B = 1(C—D). Then,

sin. Csin. D= 2 sin. (C + D) cos. (C-D), (v.) sin. C― sin. D = 2 cos. ¿(C+ D) sin. (C-D), (VI.) cos. C+cos. D = 2 cos. ¿(C+D) cos. ¿(C— D), (VII.) cos. D-cos. C 2 sin. (C+D) sin. (C-D), (VIII.)

By dividing v by VI,

sin. C+sin.D: sin.C—sin. D :: tan. †(C+D) : tan. †(C—D). (1x.)

EXERCISES.

850.-1. Demonstrate sin. 5A=5 sin. A-20 sin. A+16 sin. A. 2. Demonstrate sin. (A + B) sin. (A— B)=sin.2 A — sin.2 B.

TRIGONOMETRICAL TABLES.

851. By the application of algebra to the geometrical principles used in the construction of regular polygons, the student has found that the sine of 30° is, and the sine of 18° is (5-1). From these may be found the

cosines of these angles; then (847, IV) the sine and cosine of 15°, and then the sine of 3° (845, III). The sine of 1° may be found as follows:

sin. 3A sin. (A+2A) = sin. A cos. 2A+ cos. A sin. 2A.

=

Substituting the values of cos. 2A and sin. 2A (847),

sin. 3A: 3 cos. A sin. A― sin.3 A.

Hence (838), sin. 3A 3 sin. A-4 sin.3 A.

Put 1° for A; then, knowing the value of sin. 3°, and representing the unknown sin. 1° by x,

Only one of the roots of this equation is less than sin. 3°. It must be sin. 1o, and may be calculated by algebraic methods to any required degree of approximation.

Similarly, an equation of the fifth degree, may be formed from the value of sin. 5A; and by its means from the known sin. 1° may be found sin. 12'. Thus, by successive steps, the functions of 1' and of 1" may be found to any required degree of accuracy.

Having the sine and cosine of these small angles, the functions of their multiples may be calculated (847). This method, however, is tedious and is not used in practice. It serves to show the possibility of calculating these functions by elementary algebra and geometry. The higher analysis teaches briefer methods.

These numerical functions are called the natural sines, tangents, etc., to distinguish them from the logarithmic functions which will be defined presently.

852. The TABLE OF NATURAL SINES AND TANGENTS gives these functions to six places of figures for every 10' from 0 to 90°. It also serves as a table of cosines and cotangents.

If the sine or tangent of some intermediate angle is required, it may be found by taking a proportional part of the difference, with as much accuracy as the functions given in the table, except when the angle is nearly a right angle. For example, to find the sine 34° 23' 30", the table gives the sine of 34° 20'=.564007. Since 3' 30"

is .35 of 10', multiply 2399, the difference between this sine and that of 34° 30', by .35, and add the product to the given sine; the sum .564847 is the natural sine of 30° 23' 30".

At the beginning of this table, the functions vary with almost perfect uniformity, and in proportion to the angle. Thus, the sine and the tangent of 100' differ only by onemillionth from one hundred times the sine or the tangent of 1'. At the close of the table, the tangent varies rapidly and the sine varies slowly, and both irregularly. Therefore, for the intermediate angles (those not given in the table), the last lines are less to be relied upon than the first.

The tangent of a large angle may be found with greater accuracy by finding the cotangent of the same angle and taking its reciprocal (837).

LOGARITHMIC FUNCTIONS.

853. Before proceeding to the study of this article, the student should understand the use of the tables of logarithms of numbers.

A logarithmic sine, tangent, etc., means the logarithm. of the sine, of the tangent, etc. In the tables, the char