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 is taken as a centre, and the radius AB is equal to 1, the intercepted arc BC measures the angle A (B. III., P. XVII., S.). A B Let ABCD represent a circle whose radius is equal to 1, and AC, BD, two diameters perpendicu lar to each other. These diameters B 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 quadrant. 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 part of a right angle. 26553 5% 324000 In like manner, any angle may be expressed in terms of a right angle. 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, the angle EOB is the complement of the angle AOE, and FOB is the complement of AOF. In a right-angled triangle, the acute angles are complements of each other. B 24. The supplement of an arc is the difference between that arc and 180°. The supplement of an angle is the difference between that angle and two right angles. Thus, EC is the supplement of AE, and FC the supple ment of AF. In like manner, the angle EOC is the supple ment of the angle AOE, and FOC the supplement of AOF. In any plane triangle, any angle is the supplement of the sum of the two others. 25. Instead of the arcs themselves, certain functions of the arcs, as explained below, are used. 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 AM'. If AM is equal to M'C, AM and AM' are supplements of each other; and because MM is parallel to AC, PM is equal to P'M' (B. I., P. XXIII.): hence, the sine of an arc is equal to the sine of its supplement. M P' B N M P MNT!!! 27. The cosine of an arc is the sine of the complement of the arc, "complement sine" being contracted into cosine. Thus, NM is the cosine of AM, and NM' is the cosine of AM'. These lines are respectively equal to OP and CP'. It is evident, from the equal triangles ONM and ONM', that NM is equal to NM'; hence, the cosine of an arc is equal to the cosine of its supplement. 28. The tangent of an arc is the perpendicular to the radius at one extremity of the arc, limited by the prolongation of the diameter drawn to the other extremity. also equal. The right-angled triangles 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 tangent of AM, and AT" is the tangent of AM': hence, the tangent of an arc is equal to the tangent of its supplement. 29. The cotangent of an arc is the tangent of its complement, "complement tangent" being contracted into cotangent. Thus, BT is the cotangent of the arc AM, and BT" is the cotangent of the arc AM'. It is evident, from the equal triangles OBT' and OBT", that BT' is equal to BT"; hence, the cotangent of an are is equal to the cotangent of its supplement. When it is stated that the cotangent, tangent, &c., of an arc are equal respectively to the cotangent, tangent, &c., of its supplement, the numerical values only of the functions are referred to; no account being taken of the algebraic signs ascribed to the several functions in the different quadrants, as will be explained hereafter. The sine, cosine, tangent, and cotangent of an arc, a, 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 are 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, R denoting any radius. M a M' Let AOM represent any angle, AM an arc described from O as a centre with the radius 1, PM its sine; AM' an arc described from O as a centre, with any radius R, and P'M' its sine. Then, because OPM and OP'M' are similar triangles, we shall have, PA PA and similarly for each of the other functions: hence, 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 function of an arc whose radius is 1 multiplied by the radius R. By means of this principle, formulas may be rendered homogeneous in terms of any radius. |