107. If the given angle is greater than 90°, look for the sine, tangent, &c. of its supplement. (Art. 98, 99.) The log. sine of 96° 44' is 9.99699 of 171° 16' 9.99494 108. To find the sine, cosine, tangent, &c. of any number of degrees, minutes, and SECONDS. In the common tables, the sine, tangent, &c. are given, only to every minute of a degree.* But they may be found to seconds, by taking proportional parts of the difference of the numbers as they stand in the tables. For, within a single minute, the variations in the sine, tangent, &c. are nearly proportional to the variations in the angle. Hence, To find the sine, tangent, &c. to seconds: Take out the nuraber corresponding to the given degree and minute; and also that corresponding to the next greater minute, and find their difference. Then state this proportion; As 60, to the given number of seconds; So is the difference found, to the correction for the seconds. This correction, in the case of sines, tangents, and secants, is to be added to the number answering to the given degree and minute; but for cosines, cotangents, and cosecants, the correction is to be subtracted. For, as the sines increase, the cosines decrease. Ex. I. What is the logarithmic sine of 14° 43′ 10′′? Here it is evident, that the sine of the required angle is greater than that of 14° 43', but less than that of 14° 44'. And as the difference corresponding to a whole minute or 60" is 48; the difference for 10" must be a proportional part of 48. That is, 60': 10′′:: 48:8 the correction to be added to the sine of 14° 43′. * In the very valuable tables of Michael Taylor, the sines and tangents are given to every second. Therefore the sine of 14° 43′ 10′′ is 9.40498 2. What is the logarithmic cosine of 32° 16′ 45" ? The cosine of 32° 16' is 9.92715 Then 60": 45"::8:6 the correction to be subtract ed from the cosine of 32° 16′. Therefore the cosine of 32° 16′ 45′′ is 9.92709 The tangent of 24 15′ 18" is 9.65376 The cotangent of 31° 50' 5" The sine of The cosine of is 10.20700 58° 14′ 32′′ is 9.92956 If the given number of seconds be any even part of 60, as,,, &c. the correction may be found, by taking a like part of the difference of the numbers in the tables, without stating a proportion in form. 109. To find the degrees and minutes belonging to any given sine, tangent, &c. This is reversing the method of finding the sine, tangent, &c. (Art. 105, 6, 7.) Look in the column of the same name, for the sine, tangent, &c. which is nearest to the given one; and if the title be at the head of the column, take the degrees at the top of the table, and the minutes on the left; but if the title be at the foot of the column, take the degrees at the bottom, and the minutes on the right. Ex. 1. What is the number of degrees and minutes belonging to the logarithmic sine 9.62863? The nearest sine in the tables is 9.62865. The title of sine is at the head of the column in which these numbers are found. The degrees at the top of the page are 25, and the minutes on the left are 10. The angle required is, therefore, 25° 10' 110. To find the degrees, minutes, and SECONDS belonging to any given sine, tangent, &c. This is reversing the method of finding the sine, tangent, &c. to seconds. (Art. 108.) First find the difference between the sine, tangent, &c. next greater than the given one, and that which is next less; then the difference between this less number and the given one; Then As the difference first found, is to the other difference; So are 60 seconds, to the number of seconds, which, in the case of sines, tangents, and secants, are to be added to the degrees and minutes belonging to the least of the two num bers taken from the tables; but for cosines, cotangents, and cosecants, are to be subtracted. Ex. 1. What are the degrees, minutes, and seconds, belonging to the logarithmic sine Sine next greater 14° 44′ 9.40538 9.40498? Given sine 9.40498 Next less 14° 43′ 9.40490 Next less 9.40490 Difference 48 Difference 8 Then 48:8:: 60" : 10",. which added to 14° 43′ gives 14° 43′ 10′′ for the answer. 3. What is the angle belonging to the cosine 9.09773? Cosine next greater 82° 48′ 9.09807 Given cosine 9.09773 Difference 66 which subtracted from $2° 49', gives 82° 48′ 20′′ for the answer. It must be observed here, as in all other cases, that of the two angles, the less has the greater cosine. The angle belonging to the sin S.20621 is 9o 15'6" the tan 10.43434 is 69° 48′ 16" the cos 9.98157 16°34′30′′ the cot 10.33554 24° 47′16′′ Method of supplying the Secants and Cosecants. 111. In some trigonometrical tables, the secants and co secants are not inserted. from the sines and cosines. But they may be easily obtained For, by art. 93, proportion 3d, cos x sec=R That is, the product of the cosine and secant, is equal to the square of radius. But, in logarithms, addition takes the place of multiplication; and, in the tables of logarithmic sines, tangents, &c. the radius is 10. (Art. 103.) Therefore, in these tables, cos+sec=20. Or sec=20-cos. Again, by art. 93, proportion 6, sin x cosec=R2. Therefore, in the tables, sin+cosec=20, Or cosec=20-sin. Hence, 112. To obtain the secant, subtract the cosine from 20; and to obtain the cosecant, subtract the sine from 20. These subtractions are most easily performed, by taking the right hand figure from 10, and the others from 9, as in finding the arithmetical complement of a logarithm; (Art. 55.) observing, however, to add 10 to the index of the secant or cosecant. In fact, the secant is the arithmetical complement of the cosine, with 10 added to the index. And the ar. comp. of cos =10-cos (Art. 54.) So also the cosecant is the arithmetical complement of the sine, with 10 added to the index. The tables of secants and cosecants are, therefore, of use, in furnishing the arithmetical complement of the sine and cosine, in the following simple manner; 113. For the arithmetical complement of the sine, subtract 10 from the index of the cosecant; and for the arithmetical complement of the cosine, subtract 10 from the index of the secant. By this, we may save the trouble of taking each of the figures from 9. SECTION IIL. SOLUTIONS OF RIGHT ANGLED TRIANGLES. ART. 114. IN a triangle, there are six parts, three sides, and three angles. In every trigonometrical calculation, it is necessary that some of these should be known, to enable us to find the others. The number of parts which must be given is THREE, one of which must be a SIDE. If only two parts be given, they will be either two sides, a side and an angle, or two angles; neither of which will limit the triangle to a particular form and size. If two sides only be given, they may make any angle with each other; and may, therefore, be the sides of a thousand different triangles. Thus the two lines a and b (Fig. 7.) may belong either to the triangle ABC, or ABC', or ABC". So that it will be impossible, from knowing two of the sides of a triangle, to determine the other parts. Or, if a side and an angle only be given, the triangle will be indeterminate. Thus, if the side AB (Fig. 8.) and the angle at A be given; they may be parts either of the triangle ABC, or ABC', or AВС". Lastly, if two angles, or even if all the angles be given, they will not determine the length of the sides. For the triangles ABC, A'B'C', A"B"C", and a hundred others which might be drawn, with sides parallel to these, will all have the same angles. So that one of the parts given must always be a side. If this and any other two parts either sides or angles be known, the other three may be found, as will be shown, in this and the following section. 115. Triangles are either right angled or oblique angled. The calculations of the former are the most simple, and those which we have the most frequent occasion to make. A great portion of the problems in the mensuration of heights and distances, in surveying, navigation and astronomy, are solved by rectangular trigonometry. Any triangle whatever may be divided into two right angled triangles, by drawing a perpendicular from one of the angles to the opposite side. |