139. Sines of Angles greater than 45°. When the sines of angles up to 45° have been calculated, those of angles between 45° and 90° may be deduced by the formula sin (45° + a) sin (45° - a)=√2 sin a (Art. 45) Also, when the sines of angles up to 60° have been found, the remainder up to 90° can be found still more easily from the formula sin (60° + α) — sin (60° — α) = sin «. Having completed a table of sines, the cosines are known, since COS α = sin (90° — α). Otherwise thus: When the sines and cosines of the angles up to 45° have been obtained, those of angles between 45° and 90° are obtained from the fact that the sine of an angle is equal to the cosine of its complement, so that it is not necessary to proceed in the calculation beyond 45°. NOTE. A more modern method of calculating the sines and cosines of angles is to use series (3) and (4) of Art. 156. 140. Tables of Tangents and Secants. To form a table of tangents, we find the tangents of angles up to 45°, from the tables of sines and cosines, by means of the formula Then the tangents of angles from 45° to 90° may be obtained by means of the identity * tan (45° + α) = tan (45° − a) + 2 tan 2 α. When the tangents have been found, the cotangents are known, since the cotangent of any angle is equal to the tangent of its complement. A table of cosecants may be obtained by calculating the reciprocals of the sines; or they may be obtained more * Called Cagnoli's formula. easily from the tables of the tangents by means of the formula The secants are then known, since the secant of any angle is equal to the cosecant of its complement. 141. Formulæ of Verification. - · Formulæ used to test the accuracy of the calculated sines or cosines of angles are called Formula of Verification. It is necessary to have methods of verifying from time to time the correctness of the values of the sines and cosines of angles calculated by the preceding method, since any error made in obtaining the value of one of the functions would be repeated to the end of the work. For this purpose we may compare the value of the sine of any angle obtained by the preceding method with its value obtained independently. Thus, for example, we know that sin 18° = √5-1 (Art. 57); hence the sine of 18° may be calculated to any degree of approximation, and by comparison with the value obtained in the tables, we can judge how far we can rely upon the tables. Similarly, we may compare our results for the angles. 2210, 30°, 36°, 45°, etc., calculated by the preceding method with the sines and cosines of the same angles as obtained in Arts. 26, 27, 56, 57, 58, etc. There are, however, certain well-known formulæ of verification which can be used to verify any part of the calculated tables; these are Euler's Formulæ : sin (36° + A) sin (36° - A) + sin (72° - A) cos (36° + A) + cos (36° — A) — cos (72° + A) cos (72° — A) = cos A. Legendre's Formula: sin (54° + A) + sin (54° — A) — sin (18° + A) The verification consists in giving to A any value, and taking from the tables the sines and cosines of the angles involved: these values must satisfy the above equations. To prove Euler's Formula: sin (36° + A) sin (36° — A) = 2 cos 36° sin A. (Art. 45) sin (72° + A) — sin (72° — A) = 2 cos 72° sin A Subtracting the latter from the former, we get sin A. By substituting 90° – A for A in this formula we obtain Legendre's Formula. 142. Tables of Logarithmic Trigonometric Functions. — To save the trouble of referring twice to tables first to the table of natural functions for the value of the function, and then to a table of logarithms for the logarithm of that function it is convenient to calculate the logarithms of trigonometric functions, and arrange them in tables, called tables of logarithmic sines, cosines, etc. When tables of natural sines and cosines have been constructed, tables of logarithmic sines and cosines may be made by means of tables of ordinary logarithms, which will give the logarithm of the calculated numerical value of the sine or cosine of any angle; adding 10 to the logarithm so found we have the corresponding tabular logarithm. The logarithmic tangents may be found by the relation and thus a table of logarithmic tangents may be constructed. PROPORTIONAL PARTS. 143. The Principle of Proportional Parts. It is often necessary to find from a table of logarithms, the logarithm of a number containing more digits than are given in the table. In order to do this, we assumed, in Chapter IV., the principle of proportional parts, which is as follows: The differences between three numbers are proportional to the corresponding differences between their logarithms, provided the differences between the numbers are small compared with the numbers. By means of this principle, we are enabled to use tables of a more moderate size than would otherwise be necessary. We shall now investigate how far, and with what exceptions, the principle or rule of proportional increase is true. 144. To prove the Rule for the Table of Common Logarithms. Now let n be an integer not <10000, and d not >1; then μαζ 2n2 and μα 3n3 is not >(.0001)2, i.e., not >.0000000025; is much less than this. .. log (n + d) — log n = μ decimal places. Hence if the number be changed from n to n+d, the corresponding change in the logarithm is approximately μα n Therefore, the change of the logarithm is approximately proportional to the change of the number. If h is the circular measure of a very small angle, sin h = h nearly, and tan h = h nearly. 2 .. sin (0+ h) — sin =h cos 01 tan 0 tan If h is the circular measure of an angle not >1', then h is not>.0003 (Art. 135). sin is not > 1. h2 is not > .00000005; and 2 ..sin(+)-sin 0= cos 0, as far as seven decimal places, which proves the proposition. Similarly, sin ( − h) — sin 0 — — h cos 0, approximately. = 146. To prove the Rule for a Table of Natural Cosines. If h is the circular measure of a very small angle, sin h = h |