DEFINITIONS. FIG. 3. I. LET ABC be a plane rectilineal angle; if about B as a centre, at any distance BA, a circle ACF be described, meeting BA and BC, in A and C; the arch AC is called the measure of the angle ABC. II. The circumference of a circle is supposed to be divided into 360 equal parts called degrees, and each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds, &c. A degree is marked thus, °; a minute thus, '; a second thus, "; and so on: for example, 57° 17′ 44′′ denotes 43 degrees, 17 minutes, 44 seconds. Hence a quadrant or fourth part of the circumference contains 90°; and as many degrees, minutes, seconds, &c. as are contained in any arch, of so many degrees, minutes, seconds, &c. is the angle, of which that arch is the measure said to be; thus a right angle is said to be an angle of 90°, because its measure, a quadrant, contains 90°. COR. Whatever be the radius of the circle of which the measure of a given angle is an arch, that arch will contain the same number of degrees, minutes, seconds, &c. as is manifest from Lemma 2. III. The difference between an arch and the semi-circumference, or between an angle and 180°, is called the supplement of that arch or angle. Thus the arch CHF, (which together with AC, is equal to the semi-circumference,) is called the supplement of the arch AC; and the angle CBF, (which, together with ABC is equal to two right angles,) is called the supplement of the angle ABC. IV. The chord of an arch is a straight line drawn from one extremity of the arch to the other. Thus CA is the chord of arch CA. COR. The chord of 60° is equal to the radius of the circle; for since the angle ABC is equal to 60°, the arch AC is onesixth part of the circumference; consequently, the straight line AC is (by Cor. 15. 4.) equal to AB or BC. V. A straight line CD drawn through C, one of the extremities of the arch AC perpendicular to the diameter 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 any arch is half the chord of double that arch; for the diameter which bisects an arch also bisects the chord of that arch at right angles, (by 3. 3.) Thus CD is half CP; thus also the sine of 30° is half the radius; for it is half the chord of 60°. VI. 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. VII. A straight line AE touching the circle at A, one extremity of the arch AC, and meeting the diameter BC, passing through C, the other extremity, in E, 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. VIII. 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. 5, 7, 8, the sine, tangent, and secant of any angle It is manifest from def. 5, that CD is the sine of the angle. COR. to def. 5, 6, 7, 8. The sine, versed sine, tangent, and Fig. 4 secant, of any arch which is the measure of any given angle ABC, is to the sine, versed sine, tangent, and secant, of other arch which is the measure of the same angle, as any the radius of the first arch is to the radius of the second. Let AC, MN be measures of the angle ABC, according to def. 1. CD the sine, DA the versed sine, AE the tangent, and BE the secant of the arch AC, according to def. 5, 6, 7, 8, and NO the sine, OM the versed sine, MP the tangent, and BP the secant of the arch MN, according to the same definitions. Since CD, NO, AE, MP are parallel, CD is to NO as the radius CB to the radius NB, and AE to MP as radius AB to radius BM, and BC or BA to BD, as BN or BM to BO; and, by conversion and alternation, DA to MO as AB to MB. Hence the corollary is manifest; therefore, if the radius be supposed to be divided into any given number of equal parts, the sine, versed sine, tangent, and secant of any given angle, will each contain a given number of these parts; and, by trigonometrical tables, the length of the sine, versed sine, tangent, and secant of any angle may be found in parts of which the radius contains a given number; and, vice versa, a number expressing the length of the sine, versed sine, tangent, and secant being given, the angle of which it is the sine, versed sine, tangent and secant, may be found. IX. The difference between any angle and a right angle, is called the complement of that angle. Thus, if BH be drawn perpendicular to AB, the angle CBH will be the complement of the acute angle ABC, or of the obtuse angle CBF. In like manner, the difference between any arch and a quadrant is called the complement of that arch. Thus HC is the complement of the arch AC, or of the arch FC. X. Let HK be the tangent, CL or DB, which is equal to it, the sine, and BK the secant of CBH, the complement of ABC, according to def. 5. 7, 8, HK is called the cotangent, BD the cosine, and BK the cosecant of the angle ABC. COR. 1. The radius is a mean proportional between the tangent and cotangent of any angle ABC. For, since HK, BA are parallel, the angles HKB, ABC are equal, and KHB, BAE are right angles; therefore the triangles BAE, KHB are similar, and therefore AE is to AB, as BH or BA to HK. COR. 2. The radius is a mean proportional between the cosine and secant of any angle ABC. Since CD, AE are parallel, BD is to BC or BA, as BA to BE. Note 1.-For the sake of brevity, certain signs and characters, borrowed from arithmetic, and some obvious contractions are often used in trigonometrical investigations. Thus, if a and b denote any two numbers, their sum is denoted by a+b; their difference by a-b, or a b; their product by a axb, or a.b; their quotient by; their squares by a2 and b b2; their square roots by a and b; the square root of the sum of their squares by (a+b); the product of their sum into the sum of any other numbers c and d, by (a+b) x (c+d), or (a+b).(c+d). The mark = denotes the equality of the quantities between which it is written: thus, a=b denotes that a is equal to b; and in the statement of analogies, a:b::c:d, or a:bc:d, denotes that a is to b as c is to d, or that the ratio of a to b is the same with that of c to d. Thus also rad or R is used for radius, sin for sine, tan for tangent, sec for secant, cos for cosine, cot for cotangent, cosec for cosecant; and sin2, cos2, tan2, rad2, &c. for the squares of the sine, cosine, tangent, radius, &c. respectively. NOTE 2. In a right angled triangle, the side subtending the right angle is called the hypotenuse; and the other two sides which contain the right angle are called the legs; one of the legs is also called the perpendicular, and the other the base, according to their position. * PROP. I. FIG. 5. In a right angled plane triangle, the hypotenuse is to either of the legs as the radius to the sine of the angle opposite to that leg, and either of the legs is to the other leg as the radius to the tangent of the angle adjacent to the former leg. Let ABC be a right angled plane triangle, of which AC is the hypotenuse; assume AG as the tabular radius; from the centre A with the radius AG describe the arch DG, draw DE perpendicular to AG, and from G draw GF touching the circle in G and meeting AC in F; then is DE the sine, and FG the tangent of the arch DE, or of the angle A. The triangles AED, ABC are equiangular, because the angles AED, ABC are right angles, and the angle A is common; therefore AC is to CB as AD to DE; but AD is the radius, and DE the sine of the angle A; consequently AC: CB: rad sin A. Again, because FG touches the circle in G, AGF is a right angle, and therefore equal to the angle B, and the angle A is common to the two triangles ABC, AGF; these triangles are therefore equiangular; consequently, AB is to BC as AG to GF; but AG is the radius, and FG the tangent of the angle A; therefore AB: BC:: rad : tan A. COR. 1. Since AF is the secant of the angle A, (def. 8.), and the triangles AFG, ACB are equiangular, BA is to AC as GA to AK; that is, BA: AC:: rad: sec A. COR. 2. In a right angled plane triangle, if the hypotenuse be made radius, the sides become the sines of their opposite angles; and if either leg be made radius, the other leg becomes the tangent of its opposite angle, and the hypotenuse the secant of the same angle. PROP. II. FIG. 6, 7. The sides of a plane triangle are to one another as the sines of the angles opposite to them. In right angled triangles, this Proposition is manifest from Prop. 1; for if the hypotenuse be made radius, the sides are the sines of the angles opposite to them, and the radius is the sine of a right angle (cor. to def. 4.) which is opposite to the hypotenuse. In any oblique angled triangle ABC, any two sides AB, AC will be to one another as the sines of the angles ACB, ABC, which are opposite to them. From C, B draw CE, BD perpendicular upon the opposite sides AB, AC produced, if need be. Since CEB, CDB are right angles, BC being radius, CE is the sine of the angle CBA, and BD the sine of the angle ACB; but the two triangles CAE, DAB have each a right angle at D and E; and likewise the common angle CAB; therefore they are similar, and consequently, CA is to AB, as CE to DB; that is, the sides are as the sines of the angles opposite to them. COR. Hence of two sides, and two angles opposite to them, in a plane triangle, any three being given, the fourth is also given. *Otherwise. Fig. 16, 17. From A, draw AD perpendicular to BC; then, by Prop. 1. BA: AD:: rad: sin B. and AD AC:: sin C: rad; therefore, ex æquo inversely, BA: AC sin C: sin B. : LEMMA III. If there be two unequal magnitudes, half their difference added to half their sum is equal to the greater, and half their difference taken from half their sum is equal to the less. |