The tangent of the supplement is equal to the tangent of the arc, but with a contrary sign. Tang. BDF

BM.

The secant of the supplement is equal to the secant of the arc, but with a contrary sign. Sec. BDF= CM.

we

Signs of the functions in the four quadrants.-If divide a circle into four quadrants by a vertical and a horizontal diameter, the upper right-hand quadrant is called the first, the upper left the second, the lower left the third, and the lower right the fourth. The signs of the functions in the four quadrants are as follows:

Sine and cosecant,

Cosine and secant,

First quad. Second quad. Third quad. Fourth quad.

Tangent and cotangent,

+

+

The values of the functions are as follows for the angles specified:

+

TRIGONOMETRICAL FORMULE.

The following relations are deduced from the properties of similar triangles (Radius = 1):

sin A

cos A: sin A 1: tan A, whence tan A =

sin A

1

cos A

1

sin A

1

cot A

The sum of the square of the sine of an arc and the square of its cosine equals unity. Sin2 A+ cos2 A = 1.

Also,

1+tan2 A = sec2 A:

1+cot2 A = cosec2 4.

Functions of the sum and difference of two angles: Let the two angles be denoted by A and B, their sum A+B = C, and their difference A B by D.

sin (A+B) = sin A cos B + cos A sin B;

From these four formule by addition and subtraction we obtain

sin (A+B) + sin (A – B) = 2 sin A cos B;

sin (A — B) = 2 cos A sin B;

=

cos (A+B) + cos (A B) 2 cos A cos B;

(8)

(5)

(2)

(8)

(C + D) and B = 1⁄2(C–

(C + D) cos

Equation (9) may be enunciated thus: The sum of the sines of any two angles is equal to twice the sine of half the sum of the angles multiplied by the cosine of half their difference. These formulæ enable us to transform a sum or difference into a product.

The sum of the sines of two angles is to their difference as the tangent of half the sum of those angles is to the tangent of half their difference.

(13) The sum of the cosines of two angles is to their difference as the cotangent of half the sum of those angles is to the tangent of half their difference. cos A+ cos B 2 cos (A+B) cos 1⁄2(A – B) cot (A+B) (14) cos B-cos A 2 sin 1⁄2(A + B) sin 1⁄2(A – B) tan (4B) The sine of the sum of two angles is to the sine of their difference as the sum of the tangents of those angles is to the difference of the tangents.

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 base 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 C b

Ae, Bee 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 segments of the base may be found by Theorem III. There will then be given the hypothenuse and one side of a right-angled triangle to find the angles. For areas of triangles, see Mensuration.

ANALYTICAL GEOMETRY.

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.

Y

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.

D

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. Equation of a line passing through a given point:

y y' = a(x − x'),

in which '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 number 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:

y - y' = a(x − x').

Equation of an angle V included 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:

y22Rx - 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 "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:

A2y2+ B2x2 = A2B2,

in which A 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

2B2

2A 2B :: 2B: parameter; or parameter = A

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:

Ayy" + B2xx'' = A2В2,

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.