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To find the centre of a given circle or arc.
38. Take three points, A, B, C, any where in the circumference, or in the arc draw AB, BC; bisect these two lines by the perpendiculars, DE, FG: the point where these perpendiculars meet will be the centre sought.
The same construction serves for making a circumference pass through three given points A, B, C, and also for describing a circumference, about a given triangle.
39. In every plane triangle there are six parts: three sides and three angles. These parts are so related to each other, that if a certain number of them are known or given, the remaining ones can be determined.
40. Plane Trigonometry explains the methods of finding, by calculation, the unknown parts of a triangle when a sufficient number of the six parts is given.
It has already been shown, in the problems, that triangles may be constructed when three parts are known. But these constructions, which are called graphic methods, though perfectly correct in theory, would give only a moderate approximation in practice, on account of the imperfection of the in struments required in constructing them.
Trigonometrical methods, on the contrary, being independent of mechanical operations, give solutions with the
41. For the purposes of trigonometrical calculations, the circumference of the circle is divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes;
As the circumference of a circle may be regarded as a proper measure of angles, having their vertices at the centre, the four right angles which can be formed about the same point, are measured by 360 degrees; two right angles by 180 de grees, one right angle by 90 degrees, and an angle less than a right angle, by an arc less than 90 degrees.
Degrees, minutes, and seconds, are usually designated by the respective characters, ' Thus, 16° 12' 15" is read, 16 degrees, 12 minutes, and 15 seconds.
42. The complement of an arc is L what remains after subtracting the arc from 90°. Thus, the arc EB is the complement of AB. The sum of an arc and its complement is equal G to 90o.
43. The supplement of an arc is what remains after subtracting the arc from 180°. Thus, GF is the plement of the arc AEF. plement is equal to 180°.
The sum of an arc and its sup
44. The sine of an arc is the perpendicular let fall from one extremity of the arc on the diameter which passes through the other extremity. Thus, BD is the sine of the arc AB.
45. The cosine of an arc is the part of the diameter intercepted between the foot of the sine and centre. Thus, OD is the cosine of the arc AB.
46. The tangent of an arc is the line which touches it at one extremity, and is limited by a line drawn through the other extremity and the centre of the circle. Thus, AC is the tangent of the arc AB.
47. The secant of an arc is the line drawn from the centre of the circle through one extremity of the arc, and limited by the tangent passing through the other extremity. Thus, OC is the secant of the arc AB.
48. The four lines, BD, OD, AC, OC, depend for their values on the arc AB and the radius OA; they are thus
sin AB for
tan AB for AC
49. If ABE be equal to a quad- G rant, or 90°, then EB will be the complement of AB. Let the lines ET and IB be drawn perpendicular to OE. Then,
In general, if A is any arc or angle, we have,
cos Asin (900-A)
ET, the tangent of EB, is called the cotangent of AB;
50. If we take an arc ABEF, greater than 90°, its sine will be FH; OH will be its cosine; AQ its tangent, and OQ its secant. But FH is the sine of the arc GF, which is the supplement of AF, and OH is its cosine: hence, the sine of an arc is equal to the sine of its supplement; and the cosine of an arc is equal to the cosine of its supplement.*
Furthermore, AQ is the tangent of the arc AF, and OQ is its secant: GL is the tangent, and OL the secant, of the supplemental arc GF. But since AQ is equal to GL, and OQ to OL, it follows that, the tangent of an arc is equal to the tangent of its supplement; and the secant of an arc is equal to the secant of its supplement.*
Let us suppose, that in a circle of a given radius, the lengths of the sine, cosine, tangent, and cotangent, have been calculated for every minute or second of the quadrant, and arranged in a table; such a table is called a table of sines and tangents. If the radius of the circle is 1, the table is called a table of natural sines. A table of natural sines, therefore, shows the
* These relations are between the values of the trigonometrical lines; the
values of the sines, cosines, tangents and cotangents of all the arcs of a quadrant, divided to minutes or seconds.
If the sines, cosines, tangents and secants are known for arcs less than 90°, those for arcs which are greater can be found from them. For if an arc is less than 90°, its supplement will be greater than 90°, and the values of these lines are the same for an arc and its supplement. Thus, if we know the sine of 20o, we also know the sine of its supplement 160°; for the two are equal to each other.
TABLE OF LOGARITHMIC SINES.
51. In this table are arranged the logarithms of the numerical values of the sines, cosines, tangents and cotangents of all the arcs of a quadrant, calculated to a radius of 10,000,000,000. The logarithm of this radius is 10. In the first and last horizontal lines of each page, are written the degrees whose sines, cosines, &c. are expressed on the page. The vertical columns on the left and right, are columns of minutes.
To find, in the table, the logarithmic sine, cosine, tangent, or cotangent of any given arc or angle.
52. If the angle is less than 45o, look for the degrees in the first horizontal line of the different pages: then descend along the column of minutes, on the left of the page, till you reach the number showing the minutes: then pass along the horizontal line till you come into the column designated, sine, cosine, tangent, or cotangent, as the case may be the number so indicated is the logarithm sought. Thus, on page 37, for 19° 55′ we find,
53. If the angle is greater than 45°, search for the degrees along the bottom line of the different pages: then, ascend along the column of minutes on the right hand side of the page, till you reach the number expressing the minutes: then pass along the horizontal line into the columns designated
tang, cot, sine, or cosine, as the case may be; the number so pointed out is the logarithm required.
54. The column designated sine, at the top of the page, is designated cosine at the bottom; the one designated tang, by cotang, and the one designated cotang, by tang.
The angle found by taking the degrees at the top of the page and the minutes from the first vertical column on the left, is the complement of the angle found by taking the corresponding degrees at the bottom of the page, and the minutes traced up in the right hand column to the same horizontal line. Therefore, sine, at the top of the page, should correspond with cosine, at the bottom; cosine with sine, tang with cotang, and cotang with tang, as in the tables (Art. 49).
If the angle is greater than 90°, we have only to subtract it from 180°, and take the sine, cosine, tangent or cotangent of the remainder.
The column of the table next to the column of sines, and on the right of it, is designated by the letter D. This column is calculated in the following manner.
Opening the table at any page, as 42, the sine of 24° is found to be 9.609313; that of 24° 01', 9.609597: their difference is 284; this being divided by 60, the number of seconds in a minute, gives 4.73, which is entered in the column D, omitting the decimal point.
Now, supposing the increase of the logarithmic sine to be proportional to the increase of the arc, and it is nearly so for 60", it follows, that 473 (the last two places being regarded as decimals), is the increase of the sine for 1". Similarly, if the arc were 24° 20′ the increase of the sine for 1", would be 465, the last two places being decimals.
The same remarks are equally applicable in respect of the column D, after the column cosine, and of the column D, between the tangents and cotangents. The column D, between the columns tangents and cotangents, answers to both of these columns.
Now, if it were required to find the logarithmic sine of an arc expressed in degrees, minutes, and seconds, we have only to find the degrees and minutes as before; then, multiply the corresponding tabular number by the seconds, cut off two places to the right hand for decimals, and then add the pro