138. Angles having the Same Sine. If we let XOP = x, in this figure, and let P' be symmetric to P with respect to the axis YY', we shall have XOP'=180° x, oг π-x. And since a r sin x = sin(x) we see that x and π -x have the same sine. Furthermore, sin x = sin (360° + x), and sin (180°)=sin (360° + 180° x). That is, we may increase any angle by 360° without changing the sine. Hence we have sinx x= sin (180° — x) = sin (n · 360° + 180° — x). we may write these results as follows: sin (n. 360° + x), and Using circular measure sin x = sin (2 kπ + x), and sin (π — x) = sin (2 k + 1 π —- x). These may be simplified still more, thus: sin x = sin [nπ+(-1)"x] where n is any integer, positive or negative. Thus if n = 0 we have sin x = sin (0 · π + (− 1)ox) = sin x; if n = 1 we have sin (π- x); if n = 2 we have sin x = sin (2π + x); and so on. sin x = Since the sine is the reciprocal of the cose cant, it is evident that x and nл+(-1)x have the same cosecant. To find four angles whose sine is 0.2588, we see by the tables that sin 15° 0.2588. Hence we have sin 15° = sin [nπ + (− 1)” . 15°] = sin (π — 15°) = sin (2 π + 15°) = = sin (3 π — 15o); and so on. Exercise 69. Sines and Small Angles 1. Find four angles whose sine is 0.2756. 2. Find six angles whose sine is 0.5000. 3. Find eight angles having the same sine asπ. Given T = 3.141592653589, compute to eleven decimal places : 6. cos 1'. 7. sin 1'. 8. tan 1'. 9. sin 2'. 10. From the results of Exs. 6 and 7, and by the aid of the formula sin 2 x = 2 sin x cos x, compute sin 2', carrying the multiplication to six decimal places. Compare the result with that of Ex. 9. 11. Compute sin 1° to four decimal places. x 12. From the formula cos a = 1-2 sin2 show that cos x > 1 139. Angles having the Same Cosine. If we let - x, XOP = x, in this figure, and let P' be symmetric to P with respect to the axis XX', we shall have ▲XOP' = 360° — x, or depending on whether we think of it as a positive or a negative angle. In either case b its cosine is -, the same as cos x. b α -a+ P' Thus if n = 0, we have cos x = cos(x); if n=1, we have cos x = cos (2π±x); if n = 2, we have cos x = cos (4π + x); and so on. Since the cosine is the reciprocal of the secant, it is evident that x and 2 nπ ± x have the same secant. 140. Angles having the Same Tangent. Since we have tan x = a , we see that tan x = tan (180° + x). In where n is any integer, positive or negative. Thus if we have tan 20° given, we know that no + 20° has the same tangent. Writing both in degree measure, we may say that n. 180° + 20° has the same tangent. If n = 1, we have 200°; if n = 2, we have 380°; if n 3, we have 560°; and so on. Furthermore, if n =— -1, we have 160°; and so on. -- Since the cotangent is the reciprocal of the tangent, it is evident that x and nax have the same cotangent. Exercise 70. Angles having the Same Functions 1. Find two positive angles that have as their cosine. 2. Find two negative angles that have as their cosine. 3. Find four angles whose cosine is the same as the cosine of 25° 4. Find four angles that have 2 as their secant. 5. Find two positive angles that have 1 as their tangent. 6. Find two negative angles that have 1 as their tangent. 7. Find four angles that have √3 as their tangent. = 141. Inverse Trigonometric Functions. If y sin x, then x is the angle whose sine is y. This is expressed by the symbols x = sin-1y, or x arc sin y. In American and English books the symbol sin-1y is generally used; on the continent of Europe the symbol arc sin y is the one that is met. The symbol sin-1y is read "the inverse sine of y,' ""the antisine of y," or "the angle whose sine is y." The symbol arc sin y is read "the arc whose sine is y," or the angle whose sine is y." The symbols cos-1x, tan-1x, cot-1x, and so on are similarly used. The symbol sin-1y must not be confused with (sin y)-1. The former means the angle whose sine is y; the latter means the reciprocal of sin y. We have seen (§ 138) that sin-1 0.5000 may be 30°, 150°, 390°, 510°, and so on. In other words, there are many values for sin-1x; that is, Inverse trigonometric functions are many-valued. 142. Principal Value of an Inverse Function. The smallest positive value of an inverse function is called its principal value. For example, the principal value of sin-10.5000 is 30°; the principal value of cos-10.5000 is 60°; the principal value of tan-1(-1) is 135°; and so on. 11. Find the value of the sine of the angle whose cosine is ; 31. sin (sin-1x + sin−1y) = x √1 — y2 + y √1 − x2. 32. tan-12+ tan-1 = π. 33. 2 tan-1tan-1[2x: (1 - x2)]. -1 = 34. 2 sin-1x = sin-1(2 x √1 − x2). 35. 2 cos-1 cos1(2 x2 - 1). 1 1 1 47. tan-1 + tan-1 + tan1‡ + tan−1} = 1 π48. sin-1x + sin-1√1 - x2 = π. 49. sin-10.5+ sin-1 √3 = sin-11. 143. Graph of a Function. As in algebra, so in trigonometry, it is possible to represent a function graphically. Before taking up the subject of graphs in trigonometry a few of the simpler cases from algebra will be considered. Suppose, for example, we have the expression 3x+2. Since the value of this expression depends upon the value of x, it is called a function of x. This fact is indicated by the equation f(x)=3x+2, read "function x = 3x+2." But since f(x) is not so easily written as a single letter, it is customary to replace it by some such letter as y, writing the equation y=3x+2. If x = 0, we see that y = 2; if x =1, then y = 5; and so on. We may form a table of such values, thus: Y We may then plot the points (0, 2), (1, 5), (2, 8), · · ·, (− 1, − 1), (-2, -4),..., as in § 77, and connect them. Then we have the graph of the function 3x + 2. The graph shows that the function, y or f(x), changes in value much more rapidly than the variable, x. It also shows that the function does not become negative at the same time that the variable does, its value being 2 when x = and This kind of function in which x is of the first degree only is called a linear function because its graph is a straight line. when x = — |