PLANE TRIGONOMETRY. 20 PLANE TRIGONOMETRY is that branch of Mathematics which 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. 21. 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 equal to 1. Thus, if the vertex A be taken as a centre, and the radius AB be equal to 1, the intercepted arc BC will measure the angle A (B. III., P. XVII., S.). A4 Let ABCD represent a circle whose radius is equal to 1, and AC, BD, two diameters perpendicular to each other. These diameters divide the circumference into four equal parts, called quadrants; and because each of the angles at the centre is a right angle, it follows that a right angle is measured by a quad B D rant. An acute angle is measured by an arc less than a quadrant, and an obtuse angle, by an arc greater than a quadrant. 22. In Geometry, the unit of angular measure is a right angle; so in Trigonometry, the primary unit is a quadrant, which is the measure of a right angle. For convenience, the quadrant is divided into 90 equal parts, each of which is called a degree; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. Degrees, minutes, and seconds, are denoted by the symbols Thus, the expression 7° 22′ 33′′, is read, 7 degrees, 22 minutes, and 33 seconds. Fractional parts of a second are expressed decimally. A quadrant contains 324,000 seconds, and an arc of 7° 22' 33" contains 26553 seconds; hence, the angle measured by the latter arc, is the 55th part of a right angle. In like manner, any angle may be expressed in terms of a right angle. 24000 23. The complement of an arc is the difference between that arc and 90°. The complement of an angle is the difference between that angle and a right angle. Thus, EB is the complement of AE, and FB is the complement of AF. In like manner, EOB is the complement of AOE, and FOB is the complement of AOF In a right-angled triangle, the B F E acute angles are complements of each other. A 24. The supplement of an arc is the difference between that arc and 180°. The supplement of an angle is the dif ference between that angle and two right angles. Thus, EC is the supplement of AE, and FC the supplement of AF In like manner, EOC is the supple ment of AOE, and FOC the supplement of AOF In any plane triangle, either angle is the supplement of the sum of the other two. 25. Instead of employing the arcs themselves, we usually employ certain functions of the arcs, as explained below. A function of a quantity is something which depends upon that quantity for its value. The following functions are the only ones needed for solving triangles : 26. The sine of an arc is the distance of one extremity of the arc from the diameter, through the other extremity. Thus, PM is the sine of AM, and P'M' is the sine of If AM is equal to M'C, AM and AM' will be supplements of each other; and because MM' is parallel to AC, PM will be equal to P'M' (B. I., P. XXIII.): hence, the sine of an arc is equal to the sine of its supplement. T" B T' M' N MT A 27. The cosine of an arc is the sine of the complement of the arc. Thus, NM is the cosine of AM, and NM' is the cosine of AM'. These lines are respectively equal to OP and OP!. It is evident, from the equal triangles of the figure, that the cosine of an arc is equal to the cosine of its supple ment. 28. The tangent of an arc is the perpendicular to the radius at one extremity of the arc, limited by the prolon gation of the diameter through the other extremity Thus, AT is the tangent of the arc AM, and AT"" is the tangent of the arc AM'. T" B T M' N MT If AM is equal to M'C, AM and AM' will be supplements of each other. But AM"" and AM' are also supplements of each other: hence, the arc AM is equal to the arc AM"", and the corresponding angles, AOM and AOM"", are also equal. MT" The right-angled tri angles AOT and AOT"", have a common base AO, and the angles at the base equal; consequently, the remaining parts are respectively equal: hence, AT is equal to AT""'. But AT is the targent of AM, and AT"" is the tangent of AM': hence, the tangent of an arc is equal to the tangent of its supplement. It is to be observed that no account s taken of the alge braic signs of the cosines and tangents, the numerical values alone being referred to. 29 plement. The cotangent of an arc is the tangent of its com Thus, BT" is the cotangent of the arc AM, and BT" is the cotangent of the are AM'. The sine, cosine, tangent, and cotangent of an arc, are, for convenience, written sin a, cos a, tan a, and cot a. These functions of an arc have been defined on the supposition that the radius of the arc is equal to 1; in this case, they may also be considered as functions of the angle which the arc measures. Thus, PM, NM, AT, and BT", are respectively the sine, cosine, tangent, and cotangent of the angle AOM, as well as of the arc AM. 30. It is often convenient to use some other radius than 1; in such case, the functions of the arc, to the radius 1, may be reduced to corresponding functions, to the radius R. Let AOM represent any angle, AM an arc described from O as a centre with the radius 1, PM its sine; A'M' an arc described from 0 as a centre, with any raradius R, and P'M' its sine. Then, because OPM and OP'M' are similar triangles, we shall have, 'M' M a PA P' A' OM: PM :: OM' : P'M', or, 1: PM :: R : P'M'; whence, PM = P'M' and, P'M' PM × R; and similarly for each of the other functions. That is, 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. Also, any function of an arc whose radius is R, is equal to the corresponding func tion of an arc whose radius is 1, multiplied by the ra dius R. By making these changes in any formula, the formula will be rendered homogeneous. |