7. A heavy piece of machinery contains a bar AB with an arm AC branching from it. In order to be able to give a mechanic directions for making repairs, the owner measures off on the bar the distance AB = 10 in. and on the arm the distance AC = 10 in.; he then measures BC= 8 Make a drawing and find the angle between the arm and the bar.

4. Squared Paper. Rectangular Coördinates. It is often an advantage to draw the figure on paper ruled into squares,

AB may be laid off on one of the horizontal lines to some convenient scale, and the angles at A and B drawn by means of a protractor. From this figure, the width PR is seen immediately to be about 76 rods.

Thus, in Fig. 3, the horizontal distance from Oy to the point A is 1.4 units and the vertical distance from Ox to A is 1.5 units. To avoid confusion between points at the same distance from Ox but on opposite sides of Oy, it is customary to call distances measured to the right of Oy positive, distances measured to the left negative; thus B, Fig. 3, is said to be - 1.2 units from Oy. Similarly, distances measured downwards are called negative; for example, C in Fig. 3 is said to be .8 units from Ox.

The two distances to any point P from Ox and Oy are called the rectangular coördinates of P, and are frequently denoted by the letters x and y, respectively. The horizontal distance a is called the abscissa of P; the vertical distance y is called the ordinate of P. In giving these distances it is generally understood that the first one mentioned is x, the last y. Thus A, Fig. 3, is briefly denoted by the numbers (1.4, 1.5); B is denoted by (-1.2, 1.7); C by (−1.4, −.8); D by (2.4, −1). The lines Ox, Oy are called the axes of coördinates, or simply the axes. Ox is called the x-axis, Oy the y-axis. The point O is called the origin.

The four portions into which the plane is divided by the axes are called the first, second, third, and fourth quadrants, as in Fig. 3.

To locate a point is to describe its position in the plane in terms of its distances from the coördinate axes; e.g. (— 5, 2) is a point 5 units to the left of the y-axis and 2 units above the x-axis. To plot a point is to mark it in proper position with respect to a pair of axes.

EXERCISES II. - SQUARED PAPER

1. Locate and plot each of the following points with respect to some pair of axes:

(a) (1, 2), (b) (2, − 3), (c) (4, − 7), (d) (-5, 2), (e) (-7, -7), (ƒ) (7, 5), (g) (5, 12), (h) (8, − 3), (i) (— 5, — 5), (j) (6, − 2).

- 4) and (— 5, 4) is bisected by the

3. On what line do all the points (1, 0), (2, 0), (— 3, 0), (1.5, 0) lie? On what line do all the points (0, 0), (0, 1), (0, 2), (0, 5), (0, — 2) lie? Make a general statement about such points.

4. Find the distance from the origin to each of the points in Ex. 1, by using the folded edge of another piece of squared paper.

Compute each of the same distances by regarding each of them as the length of the hypotenuse of a right triangle, the lengths of whose sides can be read directly from the figure. Each of these methods can be used as a check on the other.

5. Find the lengths of the sides of a triangle whose vertices are (0, 0), (2, 3), (1, 5), by each of the methods of Ex. 4.

6. From the origin as one vertex construct a triangle two of whose sides are 7 and 13 units long, respectively, with an included angle of 40°. Measure, by each of the methods of Ex. 4, the length of the unknown side, and measure the angles with a protractor.

7. If a and b are any two numbers positive or negative, but not both zero, show that the line joining (a, b) and (— a, b) is bisected by the y-axis; the line joining (a, b) and (— a, —b) is bisected by the origin. 8. What is the locus of all points in the plane which have the same abscissa? The same ordinate?

9. Two objects A, B in a rectangular field are separated by a thicket, or other obstruction. To determine the distance between them, the lines AC: = 40 rods, BC = 30 rods, are measured parallel to the sides of the field. Find the distance AB, and check the answer, as in Ex. 4.

10. The positions of various objects on a rectangular farm are given by their coördinates in rods, referred to two sides of the farm as axes, as follows: house (10, 4), barn (6, 4), gate of pasture (60, 20). A railroad is constructed through the farm, passing between the house and barn, and a crossing is built at the point (3, 12). Draw a map showing the positions of the various objects. Determine how much farther it is from the house to the barn by way of the crossing than along the straight line connecting them. How much farther is it from the barn to the pasture gate by way of the crossing than along a straight line? Check each answer.

11. A certain city park is bounded by a main street, two cross streets perpendicular to it, and a stream. The distances, in feet, to the stream measured perpendicularly from the main street at 100 ft. intervals are found to be 680, 650, 525, 450, 450, 460, 540.

Draw a map of the park and determine approximately its area by counting the squares inclosed by the figure.

12. To determine the height of a tree OA standing in a level field the distance OB = 100 ft. from the base O of the tree to a point B in the field, and the angle OBA = 37°, are measured. Find approximately the height by placing the figure on squared paper, after making a preliminary estimate.

CHAPTER II

RIGHT TRIANGLES

USE OF TABLES

PART I. FUNDAMENTAL DEFINITIONS AND

PRINCIPLES

5. Tables. A method for solving right triangles that is more systematic and more accurate than the method of construction and measurement, consists essentially in making a table of the lengths of the sides and the magnitudes of the corresponding angles of all such triangles. Still the previous methods remain permanently of the utmost importance as a check.

It will be shown later that all oblique triangles can be cut up into right triangles in such a way that the same tables can be used in all cases for solving oblique triangles.

Since any triangle can be enlarged (or reduced) in size by drawing it on a larger (or smaller) scale, only the ratios of the sides are really important.

For example, it is known by geometry that if one angle of a right triangle is 30°, the side opposite this angle is one half the hypotenuse. Hence if the hypotenuse is given, that side, and hence also the other one, can be determined. If in Fig. 4, AB = 22.5, and ▲A: side BC = (1/2) (22.5) = 11.25.

30°, then the

[30°

22.5

C

B

If, for an acute angle of every right triangle, the ratio of the opposite side to the hypotenuse were known to us, then we could solve every right triangle in the same manner. Tables giving these and other ratios have been constructed.

FIG. 4

6. Definitions of the Ratios. As indicated in § 5, the ratio of two sides of a triangle does not depend upon the size of the

M

triangle, but only upon the angles. Thus in the right triangles MPN, MP'N', MP"N" of Fig. 5, in which PN, P'N', P'N'

P

P

NNN"

are perpendicular to MN, the ratios NP/MP, NP'/MP', N"P"/MP" are all equal. Moreover, if P""N"' is drawn perpendicular to MP, each of the ratios just mentioned is equal to N'"'P""' / MP"". (Why?) These ratios, then, depend only on the angle a at M. It is convenient to place the angle on a pair of axes so that the vertex falls at the origin O, one side lies along the x-axis, to the right, and the other side falls in the first quad

FIG. 5

rant. On this side take any point P at random, except O, and drop the perpendicular PM to the x-axis (see Fig. 6). Let OP= r; then by geometry

r = √x2 + y2,*

where x and y are the coördinates of the point P. The various ratios of pairs of

FIG. 6

-M

the three quantities x, y, r are the same for all points P taken in the side OP of the angle «.

(1)

(2)

(3)

У

x

These are:

called the sine of the angle a, written sin a.

called the cosine of the angle a, written cos α.

called the tangent of the angle a, written tan a. The reciprocals† of these ratios are also often used:

(4) r/y is called the cosecant of the angle a, written csc a. (5) r/x is called the secant of the angle a, written sec a. (6) x/y is called the cotangent of the angle a, written ctn α. These six ratios are collectively called trigonometric ratios or also trigonometric functions of the angle.

* The radical sign is used to denote the positive square root.

†The reciprocal of a number is unity divided by the number. The reciprocal of a common fraction is the result of inverting it; thus the reciprocal of y/r is r/y. Every number has a reciprocal except 0, which has not.