PLANE TRIGONOMETRY. III. DEFINITIONS. 1. Plane Trigonometry treats of the solution of plane triangles. In every plane triangle there are six parts, – three sides and three angles. When three of these parts are given, one being a side, the remaining parts may be found by computation. The operation of finding the unknown parts is called the solution of the triangle. 2. A plane angle is measured by the arc of a circle included between its sides; the centre of the circle being at the vertex, and its radius being 1. The circle, for convenience, is divided into 360 equal parts called degrees; 90 of these parts are included in a quadrant, which includes one-quarter of the circle, and is the measure of a right angle. Each degree is further divided into 60 equal parts called minutes, and each minute into 60 equal parts called seconds. Degrees, ininutes, and seconds are denoted by the symbols °, ', ": thus the expression 7° 22' 33" is read, 7 degrees, 22 minutes, and 33 seconds. 3. The complement of an angle is the difference between that angle and a right angle. 4. The supplement of an angle is the difference between that angle and two right angles. 11 M IN M P Р А M D 5. Instead of employing the arcs themselves, certain functions of the arcs are usually employed, as explained below. A function of a quantity is something which depends upon that quantity for its value. The sine of an angle is the distance from one extremity of the arc enclosing it, to the diameter, through the other extremity. Thus P M is the sine of the angle MO A. The cosine of an angle is the sine of the complement of the angle. Thus N M = OP is the cosine of the angle MOA. The tangent of an angle is a right line which touches the enclosing arc at one extremity, and is limited by a right line drawn from the centre of the circle through the other extremity: the sloping line which thus limits the tangent is called the secant of the angle. A T is the tangent and ( T the secant of the angle M O A. The versed sine of an angle is that part of the diameter AP which is intercepted between the foot of the sine and the extremity of the enclosing arc. The cotangent of an angle is the tangent of the complement of that angle; the co-versed sine and cosecant are similarly defined. Thus B T', BN, and OT are respectively the cotangent, co-versed sine, and cosecant of the angle M O A. These terms are in practice indicated by obvious contractions; as, sin. A for the sine of A, cos. A for the cosine of A, &c. 6. The above definitions have been made with reference to a radius of 1. Any function of an arc whose radius is R is equal to the corresponding function of an arc whose radius is 1, multiplied by the radius R. So also any function of an arc whose radius is 1 is equal to the corresponding function of an arc whose radius is R, divided by that radius. IV. NATURAL SINES, ETC. 1. Natural sines, cosines, tangents, or cotangents are those which are referred to a radius of 1. They may be used for all the purposes of trigonometrical computation; but it is found more convenient, in many cases, to employ a table of logarith 1. Logarithmic sines, cosines, tangents, or cotangents are referred to a radius of 10,000,000,000, of which the logarithm is 10. TO FIND THE LOGARITHMIC FUNCTIONS OF AN ARC WHICH IS EXPRESSED IN DEGREES AND MINUTES. 2. If the arc is less than 45°, look for the degrees at the top of the page, and for the minutes in the left-hand column; then follow the corresponding horizontal line till you come to the column designated at the top by sine, cosine, tang., or cotang., as the case may be; the number there found is the logarithm sought. Thus, log. sin. 19° 55' . log. tang, 19° 55'. 9.532312 3. If the angle is greater than 45°, look for the degrees at the bottom of the page, and for the minutes in the right-hand column; then follow the corresponding line towards the left, till you come to the column designated at the bottom by sine, cosine, tang, or cotang, as the case may be; the number there found he logarithm sought. 4. If the arc is expressed in degrees, minutes, and seconds, proceed as before with the degrees and minutes; then multiply the corresponding number taken from column D by the number of seconds, and add the product to the preceding result, for the sine or tangent, and subtract it therefrom for the cosine or cotangent. Example. 9.811952 Operation. Product 69.16 to be added . 69 Log. sine 40° 26' 28" 9.812021 5. If the arc is greater than 90°, find the required function of its supplement. Thus the logarithmic tangent of 118° 18' 25'', is equivalent to that of its supplement, or 61° 41' 35", and is 10.268732. Also the logarithmic cosine of 95° 18' 24" is 8.966080, and the log. cot. of 126° 23' 50" is 9.851619. TO FIND THE ARC CORRESPONDING TO ANY LOGARITHMIC FUNCTION. 6. This is done by a reverse process. Look in the proper column of the table for the given logarithm; if it is found there, the degrees are to be taken from the top or bottom, and the minutes from the left or right hand column, as the case may be. If the given logarithm is not found in the table, find the next less logarithm, take from the table the corresponding degrees and minutes, and set them aside. Subtract the logarithm found in the table from the given logarithm, and divide the remainder by the corresponding tabular difference. The quotient will be seconds, which must be added to the degrees and minutes set aside, in the case of a sine or tangent, and subtracted in the case of a cosine or cotangent. Example. Operation. 9.422248 .150 19/ Tabular diff. 7.68) 391(51" to be added. Hence the required arc is 15° 19' 51". . . 7. By analogous process, the arc corresponding to log. cos. VI. GENERAL PROPOSITIONS. 1. In any right-angled triangle the hypothenuse is to one of the legs as the radius to the sine of the angle opposite to that leg. And one of the legs is to the other as the radius to the tangent of the angle opposite to the latter. 2. In any plane triangle, as one of the sides is to another, so is the sine of the angle opposite to the former to the sine of the angle opposite to the latter. 3. In any plane triangle, as the sum of the sides about the vertical angle is to their difference, so is the tangent of half the sum of the angles at the base to the tangent of half their difference. 4. In any plane triangle, as the cosine of half the difference of the angles at the base is to the cosine of half their sum, so is the sum of the sides about the vertical angle to the third side, or base. Also, as the sine of half the difference of the angles at the base is to the sine of half their sum, so is the difference of the sides about the vertical angle to the third side, or base. 5. In any plane triangle, as the base is to the sum of the other two sides, so is the difference of those sides to the difference of the segments of the base made by a perpendicular let fall from the vertical angle. 6. In any plane triangle, as twice the rectangle under any two sides is to the difference of the sum of the squares of those two sides and the square of the base, so is the radius to the cosine of the angle contained by the two sides. |