ELEMENTS OF PLANE TRIGONOMETRY. T RIGONOMETRY is the application of Arithmetic to Geometry: or, more precisely, it is the application of number to express the relations of the sides and angles of triangles to one another. It therefore necessarily supposes the elementary operations of arithmetic to be understood, and it borrows from that science several of the signs or characters which peculiarly belong to it. Thus, the product of two numbers A and B, is either denoted by A.B or AXB; and the products of two or more into one, or into more than one, as of A+B into C, or of A+B into C+D, are expressed thus: (A+B) C, (A+B) (C+D), or sometimes thus, A+B × C, and A+B × C+D. The quotient of one number A, divided by another B, is written The sign is used to signify the square root: Thus M is the square root of M, or it is a number which, if multiplied into itself, will produce M. So also, ✓ M2 + N2 is the square root of M2+N2, &c. The elements of Plane Trigonometry, as laid down here, are divided into three sections; the first explains the principles; the second delivers the rules of calculation; the third contains the construction of trigonometrical tables, together with the investigation of some theorems, useful for extending trigonometry to the solution of the more difficult problems. SECTION I. LEMMA I. An angle at the centre of a circle is to four right angles as the arch on which it stands is to the whole circumference. Let ABC be an angle at the centre of the circle ACF, standing on the circumference AC: the angle ABC is to four right angles as the arch AC to the whole circumference ACF. Produce AB till it meet the circle in E, and draw DBF perpendicular to AE. Then, because ABC, ABD are two angles at the centre of the circle, ACF, the angle ABC is to the angle ABD as the arch AC to the arch AD, (33. 6.); and therefore also, the angle ABC is to four times the angle ABD as the arch AC to four times the arch AD (4. 5.). But ABD is a right angle, and therefore four times the arch AD is equal to the whole circumference ACF; therefore, the angle ABC is E H K G to four right angles as the arch AC to the whole circumference ACF. COR. Equal angles at the centres of different circles stand on arches which have the same ratio to their circumferences. For, if the angle ABC, at the centre of the circles ACE, GHK, stand on the arches AC, GH, AC is to the whole circumference of the circle ACE, as the angle ABC to four right angles; and the arch HG is to the whole circumference of the circle GHK in the same ratio. There fore, &c. DEFINITIONS. If two straight lines intersect one another in the centre of a circle, the arch of the circumference intercepted between them is called the Measure of the angle which they contain. Thus the arch AC is the measure of the angle ABC. II. If the circumference of a circle be divided into 360 equal parts, each of these parts is called a Degree; and if a degree be divided into 60 equal parts, each of these is called a minute; and if a Minute be divided into 60 equal parts, each of them is called a Second, and so And as many degrees, minutes, seconds, &c. as are in any arch, so many degrees, minutes, seconds, &c. are said to be in the angle measured by that arch. on. COR. 1. Any arch is to the whole circumference of which it is a part, as the number of degrees, and parts of a degree contained in it is to the number 360. And any angle is to four right angles as the number of degrees and parts of a degree in the arch, which is the measure of that angle, is to 360. COR. 2. Hence also, the arches which measure the same angle, whatever be the radii with which they are described, contain the same number of degrees, and parts of a degree. For the number of degrees and parts of a degree contained in each of these arches has the same ratio to the number 360, that the angle which they measure has to four right angles (Cor. Lem. 1.). The degrees, minutes, seconds, &c. contained in any arch or angle, are usually written as in this example, 49°. 36′. 24". 42′′; that is, 49 degrees, 36 minutes, 24 seconds, and 42 thirds. III. Two angles, which are together equal to two right angles, or two arches which are together equal to a semicircle, are called the Supplements of one another. IV. A straight line CD drawn through C, one of the extremities, of the arch AC, perpendicular to the di ameter passing through the other extremity A, is called the Sine of the arch AC, or of the angle ABC, of which AC is the measure. COR. 1. The sine of a quadrant, or of a right angle, is equal to the radius. COR. 2. The sine of an arch is half the chord of twice that arch: this is evident by producing the sine of any arch till it cut the circumference. V. I H H K The segment DA of the diameter passing through A, one extremity of the arch AC, between the sine CD and the point A, is called the Versed sine of the arch AC, or of the angle ABC. VI. A straight line AE touching the circle at A, one extremity of the arch AC, and meeting the diameter BC, which passes through C the other extremity, is called the Tangent of the arch AC, or of the angle ABC. COR. The tangent of half a right angle is equal to the radius. VII. The straight line BE, between the centre and the extremity of the tangent AE is called the Secant of the arch AC, or of the angle ABC. COR. to Def. 4, 6, 7, the sine, tangent, and secant of any angle ABC, are likewise the sine, tangent, and secant of its supplement CBF. |