The values of the ratios for the three angles already considered are collected in the table.

In computing the decimal fractions, we have, for example,

The first two decimal places are 0.86, but 0.87 is a better approximation to two figures.

0

Approximate values of the ratios for any acute angle ✪ may be found as follows: By means of a protractor construct the angle so that the vertex is at the origin, one side lying along the x-axis, and the other in the first quadrant. From any point P on the latter side drop the line PM perpendicular to the x-axis. Measure the lengths of the sides of the triangle OMP to find x, y, and r, and then divide y by r, x by r, and y by x. Fig. 80 simplifies and systematizes the process. It consists of a portion of a circle, whose center is the origin and whose radius is 10, and radii making angles of 10°, 20°, 20°, 30°, etc., with the x-axis.

To find the values of the ratios for = 10°, for example, we read off the coördinates of the extremity of the radius making an angle of 10° with the x-axis. They are, approximately, X = 9.8, and y 1.7, while r = 10. 10. We then have, approxi

mately

У 1.7

=

x 9.8

=

r 10

The filling in of the remainder of the table, which is to be used in the exercises below, is left as an exercise.

Ө

r

১৪

818

10° .17 .98 .17

20°

30°

40°

50°

60°

70°

80°

A method of constructing extensive tables of values of these ratios is beyond the scope of this course. But from the preceding considerations, it should be clear that, if a value of an angle 0 is given, then values of the ratios y/r, x/r, and y/x are determined. Hence these ratios are functions of (Definition, page 5).

In this chapter we shall study these functions, their reciprocals and inverses, using the graphs of the functions to tie their properties together. We shall also consider some of the important applications of the functions.

EXERCISES

1. Find the sides of a right triangle if one acute angle is 30° and the hypotenuse is 10.

2. Find the hypotenuse of an isosceles right triangle if one side is 12. 3. One acute angle of a right triangle is 60° and the leg adjacent to this angle is 25. Find the other leg and the hypotenuse.

4. Find the side of an equilateral triangle whose altitude is 8.

5. Find the side of a square if a diagonal is 30.

6. A path runs up the side of a mountain at an angle of 40° to the horizon. If a man climbs along the path for 500 yards, how high will he be above the starting point?

7. A canal makes an angle of 20° with an east and west line. If a barge moves at the rate of 4 miles an hour, how far will it move east in 5 hours? How far north?

8. Two roads cross the canal in the preceding exercise at points 200 rods apart, one running east and west, the other north and south. How large is the triangular field bounded by the canal and the roads?

9. To find the height of a tree, a line 75 feet long is paced off from the foot of the tree. At the end of the line the angle subtended by the tree is 40°. How high is the tree?

10. How high is the sun (i.e., how many degrees above the horizon) if a pole 54 feet high casts a shadow 20 feet long?

11. A balloon is anchored by a rope 1000 feet long which makes an angle of 70° with the ground. How high is the balloon? If a wrench happened to drop from the balloon, how far from the

point of anchorage would it hit the ground? 12. Find the area of a rhombus whose side

is 15 inches if one angle is 60°.

13. What is the length of the edge of a hexagonal nut that can be cut from a piece of circular stock one inch in diameter? What is the distance across the flats (i.e., between parallel edges)?

14. Holes are to be drilled through a piece of metal at the vertices of a regular hexagon, 3 inches on a side. The metal is fixed in place with a side AB along the bed of a milling machine. What displacements of the bed of the machine, in the direction of and perpendicular to AB, will bring the metal into the proper positions for drilling the holes in rotation?

A

r

B

FIG. 81.

15. An approximate geometric method of determining π is the following: Draw the diameter AB of a circle and the tangent CD at B. Construct

ZBOC equal to 30°, and make CD = 3r, where r is the radius. Then AD is approximately equal to the semicircumference πг.

Find to three decimal places the approximate value of π given by this construction.

58. Angles of any Magnitude. Let OX and OP be two lines drawn from an initial point O. Let a line start from co

nitude is determined by the amount and direction of the rotation. The numerical value of the magnitude may be given in degrees, right angles, or revolutions.

The sign is positive or negative according as the direction of rotation is counter-clockwise or clockwise, i.e., in the opposite or in the same direction as the hands of a clock rotate.

If the terminal line of a first angle is the initial line of a second the sum of the angles is defined to be the angle whose initial line is that of the first, and whose terminal line is that of the second angle. This is analogous to the sum of two lines (page 13).

Two lines determine a countless number of angles. If 0 is any one of them, the others differ from 0 by an integral multiple of 360°. They may all +n360°, where n =

be

1,

•210°

FIG. 83.

300°

represented by

2, 3,....

ㄓ

The arcs in Fig. 82 indicate the three angles:

=

0 225°, 0' 0 - 360°

=

=

135°, 0′′ = 0 + 360° = 585°. If the initial line of an angle coincides with the positive part of the x-axis, the angle is said to lie in the quadrant in which the terminal line lies. Thus in Fig. 83, the angle 300° lies in the fourth quadrant, - 210° in the second.

The positive direction on the terminal line in such a figure is defined to be away from the origin. For example, the positive direction on the terminal line of an angle of 180° is to the left.

We shall make an important use of the angles whose terminal lines bound, bisect, or trisect the four quadrants.

59. Trigonometric Functions of any Angle. Let be the number of degrees in any angle whose initial line coincides with the positive part of the x-axis, let P(x, y) be any point on the terminal line, and let OP = r.

Consider the ratio y/x, which is a negative number for the case indicated in the figure, since x = OM is negative and y = MP is positive. The numerical value of y/x may be found approximately by measuring

У

Ул

MxO

FIG. 84.

the lengths of MP and OM and dividing the former by the latter.

y

If P'(x', y') is any other point on the terminal line, the ratios y'/x' and y/x have the same sign, and also the same numerical value, since the triangles OMP and O'M'P' are similar. Hence if is given a definite value, the value of y/x is determined, and therefore the ratio y/x is a function of 0. Similarly, the ratio of any one of the numbers x, y, r, to any other is a function of the angle 0. These functions are called trigonometric functions. They are named in accordance with the definitions below, which hold for Fig. 86 A, B, C, D.

M' M

FIG. 85.