from (1). Just as angles are considered as unlimited in magnitude, so arcs are considered as unlimited in length. NOTE 1. The term circular measure is often used for radian measure, and c is used as the symbol for radians. Thus (3) is written A = 2o, B = }o. Ex. 1. Express 30° in radian measure. The term radians and the symbol for radian is usually omitted from the second members of these equations, but is always understood to be there. Ex. 2. Express 45°, 60°, 135°, 210°, 300°, 330°, 270°, 225°, — 75°, 63°, 27°, · 33o, — 150°, in radian measure, (a) as fractions of π, (b) numerically, on putting π = = 27. Ex. 3. Express the angle in degrees. means "1×2 radians." Here, "T 19 66 1r π Ex. 7. Express 1, 4, 3, 1, 1, 5, 10, 15, in degrees. 2 Measure of an arc. Since, by (5), 10' subtended circular arc = number of radians in the angle, radius then length of arc = radius × number of radians in the angle. If a denote the length of any arc AP, r the radius, ✪ the radian measure of angle AOP, then In words: The length of any circular arc is equal to the product of the radius and the radian measure of its subtended central angle. For example, the arc of 360° = 2π radii, arc of 180° = π radii, etc. These arcs are usually referred to as the arcs 2π, π, etc.; but it is always understood that the radius is the unit of measurement. The symbol, which always denotes the incommensurable number 3.14159 ..., can thus be used in three connections in trigonometry: (1) With other numbers, as a number simply. (2) With reference to angles; in which case it denotes an angle containing radians, i.e. 3.14159... radians. (3) With reference to containing 3.14159 radii. angle of radians. arcs; in which case it denotes an arc This is an arc subtended by a central The expression 180° does not mean 180° it means 180° = 3.14159 radians. = 3.14159 ...; The expression "arc π "does not mean arc 3.1416; it means "arc of 3.1416 radii." In any particular instance, the context will show to what refers, whether to angle or arc. It is evident from the second definition of radian measure that, like the trigonometric ratios, the radian measure of an angle is also a ratio of one line to another, namely, the ratio of the subtended circular arc to its radius. NOTE 2. If the radius be taken as unit length, then, by (8) or (5), the number of units of length in the arc is the same as the number of radians in the angle. EXAMPLES. 8. What is the radian measure of the angle which at the centre of a circle of radius 14 yd. subtends an arc of 8 in.? Also express the angle in degrees. Let denote the radian measure of the angle. Then = :. (4)” = 4 × 1 × 180° 8° 29′ approximately. 9. Give the trigonometric ratios of π П П π T, T, -, -π, {π. 10. Find the numerical values of (a) sin2 + cos2 &π + tan2 27 3 (b) 3 sin cos 17π tan 23 π, (c) 2 sin 23 π cos 23 π tan 1⁄23 π. 11. Find the number of radians (a) as fractions of π, (b) numerically (on putting π = 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) arc mi., 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, †, †, †, g, 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 NOTE. The values of such expressions as 2 x 4x+7, log10, 2*, 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 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, coversed sine of A= 1 − sin A, written covers A, suversed sine of 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 13, of 3 terms is 13, of 4 terms is 17, 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 − 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 |