π

10. Find the numerical values of (a) sin2 +cos2+tan2

3

(b) 3 sin cos 17 tan 23, (c) 2 sin 23 π cos 23 π tan 23 π.

11. Find the number of radians (a) as fractions of π, (b) numerically (on putting = 22), in each interior and exterior angle of the following regular polygons: pentagon, hexagon, heptagon, octagon, decagon, dodecagon, quindecagon.

12. Find the number of radians and the number of degrees in the following angles subtended at the centres of circles: (1) arc 10 in., radius 3.5 in.; (2) arc ft., radius 2 ft.; (3) arc 1 mi., radius 7920 mi.; (4) arc 250 mi., radius 8000 mi.; (5) arc 10 yd., radius 10 mi.; (6) arcmi., radius 10 ft.

13. What are the radii when an arc 10 in. in length subtends central angles containing 1, 2, 4, 6, 8, 12, 15, 20, 1, }, †, †, radians respectively ?

14. What are the radii when an arc 10 in. in length subtends central angles containing 1o, 2o, 3o, 16°, 28°, 120°, 30', 20', 10', 10", 20", 45", respectively?

15. In a circle whose radius is 10 in., what are the lengths of the arcs subtended by central angles containing 1, 4, 7, 8, 12, .5, .375, .125, radians respectively?

16. In the circle in Ex. 15, what are the lengths of the arcs subtended by central angles containing 2°, 25°, 48°, 135°, 250°, 30', 45', 30, 50′, respectively?

17. What are the areas of the circular sectors in Exs. 13, 15? [See Note C, 5.]

N.B. Questions and exercises suitable for practice and review on the subject-matter of this Chapter will be found at pages 194, 195.

CHAPTER X.

ANGLES AND TRIGONOMETRIC FUNCTIONS.

74. Chapters II., V., contain little more about the trigonometric ratios than is needed in the solution of triangles. In this and the following chapters a further study of these ratios is made. Although the results of this study are not applicable to such ordinary practical uses as the measurement of triangles, heights, and distances, yet they are very interesting in themselves, and help to give a better and fuller understanding of the connection between angles and trigonometric ratios. These results are also useful in further mathematical work, and in the study of various branches of mechanical and physical science. In reading Chapters X., XI., acquaintance will be made, or renewed, with some important general ideas of mathematics.

75. Function. Trigonometric functions. If a number is so related to one or more other numbers, that its values depend upon their values, then it is a function of these other numbers. Thus the circumference of a circle is a function of its radius; the area of a rectangle is a function of its base and height; the area of a triangle is a function of its three sides.

- 5, x2 - 4 x + 7, log10 x, 2*,

NOTE. The values of such expressions as 2 x depend upon the values given to x. These expressions are, accordingly, functions of x. A function of x is usually denoted by one of the symbols f(x), F(x), 4(x), etc., which are read "the f-function of x,' "the F-function of x," the Phi-function of x," etc.

The trigonometric ratios of an angle depend upon the value (i.e. magnitude) of the angle. On this account the trigonometric ratios are very often called the trigonometric functions. They are also frequently called the circular functions.

The trigonometric (or circular) functions include not only the

128

six functions previously discussed, namely, sine, cosine, tangent, cotangent, secant, cosecant, but also three others, viz.:

versed sine of A =

1- cos A, written vers A,

sin A, written covers A,

coversed sine of A=1

suversed sine of A=1+ cos A, written suvers A.

The versed sine is used not unfrequently; the latter two are rarely used.

It will be useful to have an idea of the meaning of the word limit as used in mathematics. In the geometrical series

the sum of 2 terms is 11, of 3 terms is 14, of 4 terms is 13, of 5 terms is 115. The sum of the series varies with the number of terms taken; and the greater the number of terms taken, the more nearly does their sum approach 2. It is stated in arithmetic and algebra that the sum of an infinitely great number of terms of this series is 1 ÷ (1 − }), i.e. 2. This simply means that, by making the number of terms as great as one please, the sum can be made to approach as nearly as one please to 2; or, in other words, the greater the number of terms taken, the more nearly does their sum approach the value 2. This idea is expressed in mathematics in slightly different language: "The limit of the sum of this series is 2." In geometry (see Note C) it is shown that if a regular polygon be inscribed in a circle, the length of the perimeter of the polygon approaches nearer and nearer to the length of the circle as the number of the sides of the polygon is increased; also the area of the polygon approaches nearer and nearer to the area of the circle. The length of the circle is said to be the limit of the length of the perimeter of the inscribed polygon, and the area of the circle is said to be the limit of the area of the polygon, as the number of its sides is indefinitely increased.

Definition. If a varying quantity approaches nearer and nearer to a fixed quantity (or given constant), so that the difference between the two quanti

ties may become, and remain, as small as one please, then the fixed quantity is called the limit of the varying quantity.

The following algebraic principles are required in some of the articles that follow :

(a) If the numerator of a fraction is finite, and its absolute value either remains constant or increases while the denominator decreases, then the absolute value of the fraction increases.

Thus, if in a either remains

-9 х

a

It is also evident that the smaller x becomes, the larger does become;

α

and that approaches an exceedingly great value when x approaches zero.

х

In other words, when the number x approaches the value zero as its limit, then the number approaches an inconceivably great value as its limit. In

a

mathematics numbers of the latter kind are each denoted by the word infinity and by the symbol ∞; that is, the symbol ∞ denotes any number which is greater than any number that can be assigned. The principle just stated, may be briefly expressed :

indicates the following reading: If x approaches zero approaches infinity as a limit. The same idea is also ex

in which the symbol

Limit a
x = 0x

α x

α

(b) It is also evident that as x increases, decreases (a remaining finite); and that when x approaches an infinitely great value, approaches zero. That is, if

х

77. Changes in the trigonometric functions as the angle increases from 0° to 360°. For convenience the revolving line will be kept constant in length in the following explanations. The student should try to deduce the changes in the functions for himself, especially after reading about the changes in the sine.

Change in sin A as A increases from 0° to 360°.

If OP be any position of the revolving line, then

Now OP is kept the same in length, say length a, as XOP increases from 0° to 360°. Hence, in order to trace changes in the sine as the angle changes, it is necessary to consider only the changes in MP. Let the angle be denoted by A.

0

а

When A 0, OP coincides with OB, and MP = 0. .. sin 0° = = : 0. As OP revolves from OX to OY, MP increases in length and is positive. When A = 90°, OP coincides with OC, and MP = a. .. sin 90° Hence, as the angle A increases from 0° to 90°, its sine increases from 0 to 1.

a

a

: 1.

= 0.

α

As OP revolves from OY to OX1, MP decreases in length and is positive. 0 When A = 180°, OP coincides with OB1, and MP= 0. :. sin 180° Hence, as the angle A increases from 90° to 180°, its sine decreases from 1 to 0.

As OP revolves from OX1 to OY1, MP increases in length and is negative; i.e. MP really decreases.