ELEMENTS OF PLANE TRIGONOMETRY.* 1. TRIGONOMETRY is the application of arithmetic to geometry, or, more precisely, it is the application of number to express the properties of angles or of circular arcs, as well as to exhibit the mutual 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 and characters which peculiarly belong to it. With these operations and characters we shall suppose the student to be acquainted. The science of Plane Trigonometry divides itself into three parts, which will be treated of in separate sections; the first contains the properties of one arc or angle; the second, those of two or more arcs or angles; and the third, those of triangles. The fourth section will exhibit the rules of trigonometrical calculation derived from the preceding; and the fifth will apply those rules. SECTION I. PROPERTIES OF ONE ARC OR ANGLE. An angle is defined in trigonometry to be the opening between two straight lines which meet one another. This definition at once indicates that the arc described about their point of intersection as a centre increases together with the angle; and the following propositions will show that the numerical value of the former correctly indicates the magnitude of the latter. 2. PROP. I. An angle at the centre of a circle is to four right angles as the arc on which it stands is to the whole circumference. Let ABC be an angle at the centre of the circle ACF, standing It may be remarked, that sections III. and IV. contain all the rules absolutely necessary for the solution of triangles; those sections, together with the definitions, are sometimes all that are studied; but this is by no means to be recommended. on the circumference AC. Draw BD at right angles to AB. Then, because ABC, ABD are two angles at the centre of the circle ACF, angle ABC: angle ABD :: arc AC: arc AD (VI. 33); therefore, also, angle ABC: 4ABD:: arc E AC: 4AD (V. 4). But 4AD is the whole circumference ACF; therefore angle ABC 4 right angles :: arc AC: whole circumference ACF. Q. E. D. 3. PROP. II. Equal angles at the centres of different circles, stand on arcs which have the same ratio to their circumferences. : Let GHK be another circle concentric with ACF; then arc AC whole circumf. ACF :: angle ABC: four right angles (Prop. 1); and arc GH whole circumference GHK angle GBH four right angles; therefore (V. 11) arc AC: whole circumference ACF:: arc GH: whole circumference GHK. Q. E.D. Hence the arcs which subtend the same angle are the same part of the whole circumference, whatever be the radii with which they are described; and, consequently, the arc is a proper measure of the angle. 4. DEFINITIONS. I. 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 these is called a second; and so on. And the number of degrees, minutes, &c., which an arc contains, is the measure of the angle subtended by it at the centre of the circle. II. If two angles together make up a right angle, the one is called the complement of the other. XIII. If two angles together make up two right angles, the one is called the supplement of the other. IV. A straight line drawn through one extremity of an arc at right angles to the diameter which passes through the other extremity, is called the sine of the arc, or of the angle which is measured by the arc. V. The portion of the diameter intercepted between the centre of the circle and the foot of the sine is called the cosine of the arc or of the angle. VI. A straight line touching the circle at one extremity of the arc, and extending to the diameter which passes through the other extremity, is called the tangent of the arc or of the angle. VII. The straight line between the centre and the extremity of the tangent is called the secant of the arc or of the angle. VIII. The segment of the diameter passing through one extre mity of an arc, which lies between the sine and that extremity, is called the versed sine of the arc or of the angle. IX. A straight line touching the circle at the distance of a quadrant from one extremity of the arc, and extending to the diameter which passes through the other extremity, is called the cotangent of the arc or of the angle. X. The straight line between the centre and the extremity of the cotangent is called the cosecant of the arc or of the angle. In the figure, ACF is a circle described about the centre B; AE, DC, BH are at right angles to AB; and KH at right angles to BH; then, to radius AB, ČD is the sine of the arc AC, or of the angle ABC; BD is its cosine; AE its tangent; BE its secant; AD its versed sine; HK its cotangent; BK its cosecant. F I H D K E 5. COR. 1. The sine of a quadrant (a quarter of the circumference), or of a right angle, is equal to the radius; and that of zero, as also of the semi-circumference, or of two right angles, is zero. The cosine of zero is equal to the radius, and that of a quadrant or of a right angle is zero. 6. COR. 2. The chord of an arc is equal to twice the sine of half the arc. For, if CD be produced to meet the circumference, the chord will be bisected in D (III. 3); and the angle ABC will be half the angle which the chord subtends at the centre (I. 8); and, consequently, the arc AC will be half the arc cut off by the chord (III. 26). 7. COR. 3. Hence (III. 29) the sines of equal arcs are equal to one another. 8. The word function is sometimes employed to express any of the trigonometrical lines; thus, the function of an arc or angle is its sine, or cosine, or tangent, &c., as the case may be. It is evident that when the angle is less than a right angle, the cofunction of the angle is the function of its complement. 9. PROP. III. To express the relations which exist between the sine and cosine of an angle, and those of its complement, supplement, &c. Let A denote the angle BAC, expressed in degrees, of which the complement is CAD. Produce BA, CA, DA to E, F, G; make AF AC, and draw CB, CD, FE, FG at right angles to EB, DG respectively. Then E the triangles CAB, FAE are equal in every respect (I. 26), and CB=FE, AB-AE. Similarly, CD=FG and AD=AG. And because DB, EG are parallelograms (I. 28), AB =DC, &c. (I. 34). Hence, to radius AC, A B cos (90-A)=cos CAD-AD=CB=sin CAB=sin A; cos (90+A)=cos CAG=AD=CB=sin A*; sin (90+A)=CD=AB=cos A; sin (180-A)=sin CAE=CB=sin A; cos (180-A)=AB*=cos A*; sin (180+A)=sin BAF, i. e. sin (BAD+DAF)=FE=CB*=sin A*; cos (180+A)=AE=AB*=cos A* ; whence it appears that if an angle be added to or taken from two right angles, the function (sine or cosine) of the sum or difference is the same (abstracting from sign) as that of the angle itself; but if an angle be added to, or taken from one right angle, the function of the sum or difference is the complementary function of the angle itself. In the same way it may be proved that if an angle be added to or taken from any even number of right angles, the function (sine or cosine) of the sum or difference is the same as that of the angle itself; but if an angle be added to or taken from any odd number of right angles, the function of the sum or difference is the complementary function of the angle itself. 10. PROP. IV. To express the relations which exist between the different functions of the same angle. Let the angle ABC be denoted by A. Retaining the figure and construction of Art. 4, we obtain, by similar triangles, AE: AB:: CD: BD (VI. 4), therefore (Art. 4), tan A: R:: sin A: cos A; (1) BE: BA:: BC: BD, therefore, sec A: R:: R: cos A; (2) HK: BH:: CL: BL :: BD: CD, therefore, cot A: R:: cos A: sin A; (3) HK: BH::AB: AE, therefore, cot A:R::R:tan A ; (4) BK: BH:: BC: CD, therefore, cosec A: R:: R: sin A; (5). H K By (I. 47) DC2+BD2=BC2; therefore, sin 2A+cos 2A=R2 (6); BA2+AE2=BE2; therefore, R 2+tan 2A=sec2A (7); BH2+HK2BK2; therefore, R2+cot 2A=cosec 2A (8). 11. COR. By means of these eight relations we can determine the properties of all the trigonometrical functions, when we know those of the sine and cosine. For example, if B=90+A; by (1) tan B R sin B: cos B; but sin B and cos B are (Art. 9) respectively cos A* and sin A; therefore, tan B: R:: cos A*: • Abstracting from its sign, which is . See Art. 18. : sin A; but cot A: R cos A: sin A (by 3); therefore, tan B*-cot A. Again, if C-180-A, tan C: R:: sin C cos C by (1) sin A cos A* (Art. 9); therefore, tan C*-tan A. : The same may be shown of all the other trigonometrical functions; hence, the conclusion of Art. 9 is not confined to the sine and cosine, but applies to them all. Many writers on trigonometry define only the sine and cosine from the construction, regarding the other trigonometrical functions simply as lines which possess the properties exhibited by the proportions given in Art. 10. This method has some advantages. 12. PROP. V. The trigonometrical functions of the same angle to different radii are to one another respectively as the radii. Let BC, BN be the radii of the circles AC, MN; then CD, NO are the sines of the angle ABC to these radii respectively; BD, BO the cosines; AE, MP the tangents; BE, BP the secants; AD, MO the versed sines. By similar triangles (VI. 4); B CD: NO:: BC: BN, AE: MP:: BA: BM, BE: BP: BA: BM, Also, BC: BD :: BN: BO, Therefore, by conversion, BC: AD :: BN: MO, And, alternately, BC: BN :: AD: MO, P OM D A E which are the propositions to be proved. And therefore, if tables be constructed exhibiting, in numbers, the sines, tangents, secants, and versed sines of certain angles to a given radiús, they will exhibit the ratios of these functions of the same angles to any radius whatever. In such tables, which are called trigonometrical tables, the radius is generally supposed to be either 1, or the tenth power of 10. In arithmetical computations it is more convenient to suppose it to be 1, because, when it appears as a multiplier, it may then be omitted altogether. We shall consequently adopt this value of the radius in all our arithmetical calculations. 13. PROP. VI. To find the arithmetical relations between the different functions of the same angle. The arithmetical expression for a rectangle is the product of the numbers which represent the containing lines (VI. 23). Now, * Abstracting from the sign, which is negative, Art 18. |