Solution of Plane Right-angled Triangles. Let A and B be the two acute angles and C the right angle, and a, b, and c the sides opposite these angles, respectively, then we have 1. In any plane right-angled triangle the sine of either of the acute angles is equal to the quotient of the opposite leg divided by the hypothenuse. 2. The cosine of either of the acute angles is equal to the quotient of the adjacent leg divided by the hypothenuse. 3. The tangent of either of the acute angles is equal to the quotient of the opposite leg divided by the adjacent leg. 4. The cotangent of either of the acute angles is equal to the quotient of the adjacent leg divided by the opposite leg. 5. The square of the hypothenuse equals the sum of the squares of the other two sides. Solution of Oblique-angled Triangles. The following propositions are proved in works on plane trigonometry. In any plane triangle Theorem 1. The sines of the angles are proportional to the opposite sides. Theorem 2. The sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their differ ence. Theorem 3. If from any angle of a triangle a perpendicular be drawn to the opposite side or base, the whole hase will be to the sum of the other two sides as the difference of those two sides is to the difference of the segments of the base. CASE I. Given two angles and a side, to find the third angle and the other two sides. 1. The third angle = 180° - sum of the two angles. 2. The sides may be found by the following proportion : The sine of the angle opposite the given side is to the sine of the angle opposite the required side as the given side is to the required side. CASE II. Given two sides and an angle opposite one of them, to find the third side and the remaining angles. The side opposite the given angle is to the side opposite the required angle as the sine of the given angle is to the sine of the required angle. The third angle is found by subtracting the sum of the other two from 180°, and the third side is found as in Case I. CASE III. Given two sides and the included angle, to find the third side and the remaining angles. The sum of the required angles is found by subtracting the given angle from 180°. The difference of the required angles is then found by Theorem II. Half the difference added to half the sum gives the greater angle, and half the difference subtracted from half the sum gives the less angle. The third side is then found by Theorem I. Another method : Given the sides c, b, and the included angle A, to find the remaining side a and the remaining angles B and C. From either of the unknown angles, as B, draw a perpendicular B e to the opposite side. Then Ae = c cos A, Be c sin A, e Cb- Ae, Be÷e C tan C. Or, in other words, solve Be, Ae and Be Cas right-angled triangles. CASE IV. Given the three sides, to find the angles. Let fall a perpendicular upon the longest side from the opposite angle, dividing the given triangle into two right-angled triangles. The two seg ments of the base may be found by Theorem III. There will then be given the bypothenuse and one side of a right-angled triangle to find the angles. For areas of triangles, see Mensuration. Triangles. on plane trigonometry. In Tonal to the opposite sides. - difference as the tangent ngent of half their differ erpendicular be drawn to the sum of the other two ifference of the segments third angle and the other two angles. 2. The sides the sine of the angle opquired side. one of them, to find the posite the required angle quired angle. the other two from 180°, o find the third side and acting the given angle hen found by Theorem the greater angle, and ves the less angle. The d the remaining side a erpendicular Be to the ee C tan C. angled triangles. the opposite angle, ngles. The two seg ere will then be given to find the angles. ANALYTICAL GEOMETRY. 69 ANALYTICAL GEOMETRY. C Y Analytical geometry is that branch of Mathematics which has for its object the determination of the forms and magnitudes of geometrical magnitudes by means of analysis. Ordinates and abscissas.--In analytical geometry two intersecting lines YY', XX' are used as coördinate axes, XX' being the axis of abscissas or axis of X, and YY' the axis of ordinates or axis of Y. A, the intersection, is called the origin of coördinates. The distance of any point P from the axis of Y measured parallel to the axis of X is called the abscissa of the point, as AD or CP, Fig. 71. Its distance from the axis of X, measured parallel to the axis of Y, is called the ordinate, as AC or PD. The abscissa and ordinate taken together are called the coördinates of the point P. The angle of intersection is usually taken as a right angle, in which case the axes of X and Y are called rectangular coördinates. x A FIG. 71. The abscissa of a point is designated by the letter x and the ordinate by y. The equations of a point are the equations which express the distances of the point from the axis. Thus xa, y = b are the equations of the point P. Equations referred to rectangular coördinates.-The equation of a line expresses the relation which exists between the coördinates of every point of the line. Equation of a straight line, y = ax + b, in which a is the tangent of the angle the line makes with the axis of X, and b the distance above A in which the line cuts the axis of Y. Every equation of the first degree between two variables is the equation of a straight line, as Ay+Bx + C = 0, which can be reduced to the form y = ax ± b. Equation of the distance between two points: D= √(x" - x')2 + (y'' — y')', in which x'y', x''y" are the coördinates of the two points. in which x'y' are the coördinates of the given point, a, the tangent of the angle the line makes with the axis of x, being undetermined, since any num ber of lines may be drawn through a given point. Equation of a line passing through two given points: Equation of a line parallel to a given line and through a given point: Equation of an angle Vincluded between two given lines: in which a and a' are the tangents of the angles the lines make with the axis of abscissas. If the lines are at right angles to each other tang V = ∞, and 1 + a'a = 0. Equation of an intersection of two lines, whose equations are Equation of a perpendicular from a given point to a given line: The circle.-Equation of a circle, the origin of coördinates being at the centre, and radius = R: x2 + y2 = R2. If the origin is at the left extremity of the diameter, on the axis of X: y2 2Rx x2. If the origin is at any point, and the coördinates of the centre are x'y': (x − x')2+(y — y')2 = R2. Equation of a tangent to a circle, the coördinates of the point of tangency being a"y" and the origin at the centre, yy" + xx" = R2. The ellipse.-Equation of an ellipse, referred to rectangular coördinates with axis at the centre: in which 4 is half the transverse axis and B half the conjugate axis. Equation of the ellipse when the origin is at the vertex of the transverse axis: The eccentricity of an ellipse is the distance from the centre to either focus, divided by the semi-transverse axis, or The parameter of an ellipse is the double ordinate passing through the focus. It is a third proportional to the transverse axis and its conjugate, or Any ordinate of a circle circumscribing an ellipse is to the corresponding ordinate of the ellipse as the semi-transverse axis to the semi-conjugate. Any ordinate of a circle inscribed in an ellipse is to the corresponding ordinate of the ellipse as the semi-conjugate axis to the semi-transverse. Equation of the tangent to an ellipse, origin of axes at the centre: y''x" being the coördinates of the point of tangency. Equation of the normal, passing through the point of tangency, and perpendicular to the tangent: The normal bisects the angle of the two lines drawn from the point of tangency to the foci. The lines drawn from the foci make equal angles with the tangent. The parabola.-Equation of the parabola referred to rectangular coördinates, the origin being at the vertex of its axis, y2 = 2px, in which 2p is the parameter or double ordinate through the focus. The parameter is a third proportional to any abscissa and its corresponding ordinate, or Equation of the tangent: xy::y:2p. The sub-normal, or projection of the normal on the axis, is constant,.and equal to half the parameter. The tangent at any point makes equal angles with the axis and with the line drawn from the point of tangency to the focus. The hyperbola.-Equation of the hyperbola referred to rectangular coördinates, origin at the centre: in which 4 is the semi-transverse axis and B the semi-conjugate axis. Equation when the origin is at the right vertex of the transverse axis: B2 Conjugate and equilateral hyperbolas.-If on the conjugate axis, as a transverse, and a focal distance equal to √A2 + B2, we construct the two branches of a hyperbola, the two hyperbolas thus constructed are called conjugate hyperbolas. If the transverse and conjugate axes are equal, the hyperbolas are called equilateral, in which case y2 — x2 = — A2 when A is the transverse axis, and x2 - - y2: B2 when B is the trans verse axis. The parameter of the transverse axis is a third proportional to the transverse axis and its conjugate. 2A 2B: 2B: parameter. The tangent to a hyperbola bisects the angle of the two lines drawn from the point of tangency to the foci. The asymptotes of a hyperbola are the diagonals of the rectangle described on the axes, indefinitely produced in both directions. In an equilateral hyperbola the asymptotes make equal angles with the transverse axis, and are at right angles to each other. The asymptotes continually approach the hyperbola, and become tangent to it at an infinite distance from the centre. Conic sections.-Every equation of the second degree between two variables will represent either a circle, an ellipse, a parabola or a hyperbola. These curves are those which are obtained by intersecting the surface of a cone by planes, and for this reason they are called conic sections. Logarithmic curve.-A logarithmic curve is one in which one of the coordinates of any point is the logarithm of the other. The coördinate axis to which the lines denoting the logarithms are parallel is called the axis of logarithms, and the other the axis of numbers, If y is the axis of logarithms and x the axis of numbers, the equation of the curve is y = log x. If the base of a system of logarithms is a, we have a = x, in which y is the logarithm of x. Each system of logarithms will give a different logarithmic curve. If y = 0,1. Hence every logarithmic curve will intersect the axis of numbers at a distance from the origin equal to 1. DIFFERENTIAL CALCULUS. The differential of a variable quantity is the difference between any two of its consecutive values; hence it is indefinitely small. It is expressed by writing d before the quantity, as dx, which is read differential of x. is called the differential coefficient of y regarded as a func The differential of a function is equal to its differential coefficient mul dy tiplied by the differential of the independent variable; thus, dx = - dy. dx The limit of a variable quantity is that value to which it continually approaches, so as at last to differ from it by less than any assignable quantity. The differential coefficient is the limit of the ratio of the increment of the independent variable to the increment of the function. The differential of a constant quantity is equal to 0. The differential of a product of a constant by a variable is equal to the constant multiplied by the differential of the variable. dy If u = Av, du = Adv. In any curve whose equation is y = f(x), the differential coefficient = tan a; hence, the rate of increase of the function, or the ascension of the curve at any point, is equal to the tangent of the angle which the tangent line makes with the axis of abscissas. dx All the operations of the Differential Calculus comprise but two objects: 1. To find the rate of change in a function when it passes from one state of value to another, consecutive with it. 2. To find the actual change in the function: The rate of change is the differential coefficient, and the actual change the differential. Differentials of algebraic functions.-The differential of the sum or difference of any number of functions, dependent on the same variable, is equal to the sum or difference of their differentials taken separately: = If u = y + zw, du dy + dz - dw. The differential of a product of two functions dependent on the same variable is equal to the sum of the products of each by the differential of the other: d(uv) du dv + uv บ и The differential of the product of any number of functions is equal to the sum of the products which arise by multiplying the differential of each function by the product of all the others: d(uts) = tsdu+usdt + utds. The differential of a fraction equals the denominator into the differential of the numerator minus the numerator into the differential of the denominator, divided by the square of the denominator: If the numerator is constant, du = 0, and dt = The differential of the square root of a quantity is equal to the differential of the quantity divided by twice the square root of the quantity: udv v2 |