But the parallelograms CA, CD, being equiangular, are as the rectangles of the sides which contain the equal angles (B. IV., Pr. 24, Cor. 2); hence the parallelogram CD is equal to the parallelogram CA. EXERCISES ON THE HYPERBOLA. 1. In an hyperbola, the tangents at the vertices of the transverse axis will meet the asymptotes in the circumference of the circle described on FF, as a diameter. 2. If DM be drawn parallel to CG (fig., Pr. 14), meeting the transverse axis in M, then ME=BC. 3. If an hyperbola and an ellipse have the same foci, they cut one another at right angles. 4. If DG (fig. 2d, Pr. 11) be the ordinate of a point D, and GK be drawn parallel to AD to meet CD in K, then AK is parallel to the tangent at D.. 5. If from any point of the hyperbola lines be drawn parallel to, and terminating in the asymptotes, the parallelogram so formed will be equal to one eighth of the rectangle described on the axes. 6. An ordinate to the transverse axis of an hyperbola is 43 inches, and the corresponding abscissas are 30 and 85 inches; required the latus rectum. 7. If the axes of an hyperbola are 65 and 54 inches, what is the radius of a circle described to touch the curve, when its centre is in the transverse axis produced, at the distance of 112 inches from the centre of the hyperbola? 8. If the axes of an hyperbola are 65 and 54 inches, what is its latus rectum, and what is the position of its directrix? 9. The conjugate axis of an hyperbola is 52 inches, the latus rectum 42 inches, and an ordinate of 36 inches is drawn to the transverse axis; determine where the tangent line drawn through the extremity of this ordinate meets the transverse axis. 10. Determine where the tangent line in the last example meets the conjugate axis. PLANE TRIGONOMETRY. 1. TRIGONOMETRY is that branch of Mathematics which teach es how to determine the several parts of a triangle by means of others that are given. In a more enlarged sense, it embraces the investigation of the relations of angles in general. Plane Trigonometry treats of plane angles and triangles; Spherical Trigonometry treats of spherical triangles. 2. In every triangle there are six parts: three sides and three angles. These parts are so related to each other that when any three of them are given, provided one of them is a side, the remaining parts can be determined. 3. In order to subject angles to computation, they must be expressed by numbers. The units by which angles are expressed are the degree, minute, and second, designated by the characters °,',". A degree is the 90th part of a right angle, or the 360th part of the whole angular space about a point. A right angle is expressed by 90°; two right angles by 180°; and the whole angular space about a point by 360°. A minute is an angle equal to the 60th part of a degree. Therefore one degree=60'. A second is an angle equal to the 60th part of a minute. Therefore one minute=60′′. Angles less than a second are expressed as decimal parts of a second. Thus 4th of four right angles will be expressed by 51° 25' 42."86. 4. Sinee angles at the centre of a circle are proportional to the arcs intercepted between their sides, these arcs may be taken as the measures of the angles. An angle may therefore be measured by the number of units of arc intercepted on the circumference. The units of arc are also the degree, minute, and second. They are the arcs which subtend angles of a degree, a minute, and a second respectively at the centre. The quadrant is therefore expressed by 90°; the semi-circumference by 180°; and the whole circumference by 360°. The radius of the circle employed in measuring angles is arbi trary, and, for convenience, is generally taken as unity. When this is not done, it is denoted by its initial letter R. 5. The circumference of a circle whose diameter is unity is 3.14159. If the radius be unity, the semi-circumference, or an arc of 180°, will be 3.14159. Hence the length of an arc of 1o 0.01745; and the length of an arc of 1' 0.00029, etc. will be will be 6. The complement of an arc or angle is the remainder obtained by subtracting the arc or angle from 90°. Thus the complement of 25° 15' is 64° 45'. Since the two acute angles of a right-angled triangle are together equal to a right angle, each of them must be the complement of the other. In general, if we represent any arc by A, its complement is 90°-A. Hence, if an arc exceeds 90°, its complement must be negative. Thus the complement of 113° 15' is -23° 15'. See Art. 79. 7. The supplement of an arc or angle is the remainder obtained by subtracting the arc or angle from 180°. Thus the supplement of 25° 15' is 154° 45'. Since in every plane triangle the sum of the three angles is 180°, either angle is the supplement of the sum of the other two.. In general, if we represent any arc by A, its supplement is 180°-A. Hence, if an arc is greater than 180°, its supplement must be negative. Thus the supplement of 200° is —20°. 8. The sine of an arc is the perpendicular let fall from one extremity of the arc upon the diameter passing through the other extremity.. D Cotan. L 1 Cosine K •Sec. I Sine Tangent C B Cos. Vers. E M A Thus FG is the sine of the arc AF, or of the angle ACF. Every sine is half the chord of double Thus the sine FG is the half chord of the arc The chord which the arc. 9. The tangent of an arc is the line which touches the circle at one extremity of the arc, and is limited by a line drawn from the centre through the other extremity. Thus AI is the tangent of the arc AF, or of the angle ACF. 10. The secant of an arc is the line drawn from the centre of the circle through one extremity of the arc, and is limited by the tangent drawn through the other extremity. Thus CI is the secant of the arc AF, or of the angle ACF. In the preceding definitions of sine, tangent, and secant, the ra dius of the circle has been assumed as unity. In a circle of any other radius, we must suppose these lines to be divided by that radius. 11. The cosine of an arc is the sine of the complement of that arc. Thus the arc DF, being the complement of AF, FK, or its equal CG, is the sine of the arc DF, or the cosine of the arc AF. The cotangent of an arc is the tangent of the complement of that arc. Thus DL is the tangent of the arc DF, or the cotangent of the arc AF. The cosecant of an arc is the secant of the complement of that arc. Thus CL is the secant of the arc DF, or the cosecant of the arc AF. In general, if we represent any angle by A, . cos. A sine (90°-A); cosec. A sec. (90°—A). Since in a right-angled triangle either of the acute angles is the complement of the other, the sine, tangent, and secant of one of these angles is the cosine, cotangent, and cosecant of the other. 12. The versed sine of an arc is that part of the diameter intercepted between the extremity of the arc and the foot of the sine. Thus GA is the versed sine of the arc AF, or of the angle ACF. The versed sine of an acute angle ACF is equal to the radius : minus the cosine CG. The versed sine of an obtuse angle BCF is equal to radius plus the cosine CG; that is, to BG. 13. The sine, tangent, and secant of any arc are equal to the sine, tangent, and secant of its supplement. Thus FG is the sine of the arc AF, or of its supplement BDF. AI, the tangent of the arc AF, is equal to BM, the tangent of the arc BDF. And CI, the secant of the arc AF, is equal to CM, the secant of the arc BDF. 14. Fundamental formula. The relations of the sine, cosine, etc., to each other may be derived from the proportions of the M 3. GF: CG:: CD: DL; that is, sin. A: cos. A::R: cot. A. 4. GF: CF:: CD: CL; that is, sin. A: R::R: cosec. A. 5. AI: AC:: CD: DL; that is, tang. A: R::R: cot. A. The preceding formula will be frequently referred to hereafter. 15. Given the sine of an angle, to find the cosine, tangent, etc. In the right-angled triangle CGF, we find CG2+GF2=CF2; that is, sin.2A+cos.2A=R2, where sin.2A signifies "the square of the sine of A." When radius is taken as unity, we have cos. A=√1-sin.2A=√(1+sin. A) (1—sin. A). When the sine and cosine of an ångle have been determined, the tangent may be found by Eq. 1, Art. 14, . |