PLANE TRIGONOMETRY. CHAPTER I. ON THE METHOD OF REFRESENTING LINES AND ANGLES, 1. THE magnitudes of lines may be represented by algebraical quantities. To measure a line we find how many times it contains a fixed and definite line, as an inch or a yard, which has been previously fixed on as the unit of length. Thus, if an inch be taken as the unit of length, we say that a line is 30, if it contain an inch thirty times; and in like manner we call a line a, if it contain the unit of length a times. 2. If lines drawn in a given direction from a fixed line be represented by positive quantities, then those which are drawn from that line in a contrary direction will be represented by negative quantities. 37 If to a line AB measured from A towards the right of the fixed line Ay it be required to add a given line, it is evident that AB A must be produced to a point C such that BC shall be equal to that given line, and if from AB we have to take a given line, we must cut off from BA a part BC' equal to the given line, and AC' will be the required remainder. Let AB be represented by a, and BC by b, and make Now if b be greater than a, C' lies on the other side of A. And AC' = a b = (ba), which is a negative quantity = equal in magnitude to the dif ference of the lines b and a; which difference we see lies in c this case to the left of Ay. Hence, if a line be represented by a negative quantity, it is meant that it is measured from Ay in a direction contrary to that in which the lines represented by positive quantities were measured. 3. Let XAX', YAY' be two fixed lines at right angles to each other, and produced, if necessary, indefinitely. Then the position of any point P, in the plane of these lines is known, X if we know the magnitudes of the perpendiculars let fall from P1 upon the lines XAX' and YAY', viz. PN, and PM (or AN1, which is equal to PM). P M N N2 A NA N X Y AN1, N1P1, are called the co-ordinates of P、 referred to the rectangular axes AX and AY. With respect to lines measured from A in the direction of the axis XAX', it is usual to call those positive which are measured to the right of the axis YAY', and therefore those will be negative which are measured to the left of YAY'. With respect to lines measured in the direction of YAY' it is usual to call those positive which lie on the upper side of XAX', and therefore those will be negative which lie on the lower side of XAX'. Thus if P, be a point in the space contained by AX and AY, its ordinate AN, which is measured in the direction of XAX' is positive, because it lies to the right of YAY'; and its ordinate N1P1 measured in the direction of YAY' is positive, because it lies on the upper side of XAX'. But 2 if P2 be the point, its ordinate AN, measured in the direction of XAX' is negative, because it lies to the left of YAY'; and the ordinate N, P, measured in the direction of YAY' is positive, because it lies on the upper side of XAX'. 2 Similarly, the ordinates of P and P1 measured in the direction of XAX' are negative and positive respectively; and their ordinates measured in the direction of YAY' are negative in both cases. D C E' 4. DEF. If a straight line revolve in one plane round its extremity A from a given position as AB into any other position as AC, the inclination of AC to AB is called an angle (4); and the angle is signified by the letters BAC or CAB, the middle letter being that placed at the point in which the two lines meet. B' By continuing this revolving motion, we may suppose the angle to become of any magnitude whatever. 5. DEF. If AD be equally inclined to the parts AB, AB' of the straight line BB', each of the angles BAD, B'AD is called a right angle. 6. DEF. An acute angle is less, and an obtuse angle is greater, than a right angle. 7. If the angles formed by AC revolving in one direction, as BCD, from AB be considered positive, then if AC revolve in the contrary direction from AB it will form negative angles. If to the angle BAC, Fig. Art. 4, it be required to add a given angle, CA must move in the direction BCD through an angle CAE equal to the given angle, and the whole BAE will be the angle required. And if it be required to take a given angle from BAC, CA must evidently move in the contrary direction till it come into a position E'A, such that < CAE' is equal to the angle to be subtracted. Then CAE + LE'AB = L BAC; . LEABL BAC CAE'. B a negative quantity, whose magnitude is the difference between the angles CAE' and BAC; which difference, we see, lies in this case on the lower side of AB. Hence, by calling an angle negative, we mean that it is formed by the revolving line moving in a direction contrary to that in which it revolved to trace out positive angles. 8. A right angle is divided by the English into 90 equal angles which are called degrees; a degree into 60 minutes; a minute into 60 seconds; a second into 60 thirds; and so on to fourths, fifths, &c. And in the same manner as the length of a given line is determined by finding how many times it contains the unit of length, (as a yard), and its subdivisions, (viz. feet and inches)-so the magnitude of an angle is determined by finding how many times it contains a degree and its subdivisions. A degree and its subdivisions are thus marked, 24o, 50′, 34", 42""; which denotes an angle containing 24 degrees, 50 minutes, 34 seconds, and 42 thirds. In practice the subdivisions of the angle beyond seconds are not commonly used; if very great accuracy be required, the subdivisions beyond seconds are expressed in decimal parts of a second: thus since. the angle above might have been written 24, 50′, 34′′7. 9. By the French and other Continental Mathematicians, a right angle is divided into 100 equal angles called grades; a grade into 100 minutes; a minute into 100 seconds; and so on; and the divisions are thus marked, 269, 24, 32", 47"". The above angle might have been written 269.243247. Whence it appears that if the French division be adopted, arithmetical operations can be performed on angles in the same manner as on any other decimal fractions; an advantage which does not attend the English division of the angle. 10. To find the relation between E and F, the number of degrees and grades contained in the same angle BAC. (Art. 14. Fig. 1.) |