= 94. From the equation y sin o, we know that is an angle whose sine is y. The last statement is expressed by the notation + sin 1y. = Hence sin-ly is an angle. sin-ly is sometimes read, "an angle whose sine is y," sometimes "arc-sine y," but more frequently "anti-sine y." But it must be remembered that it means "an angle whose sine is y." CHAPTER XII INVERSE TRIGONOMETRIC FUNCTIONS 99 cos-ly means "an angle whose cosine is y." tan-ly means "an angle whose tangent is y. csc-ly means "an angle whose cosecant is y." sec-ly means "an angle whose secant is y." cot-ly means "an angle whose cotangent is y." These are read "anti-cosine y," "anti-tangent y," "anti-cosecant y," "anti-secant y," "anti-cotangent y," respectively. EXAMPLE. 30° = sin = 45° = 95. The expressions sin-ly, cos-1y, etc., are called the Inverse Trigonometric Functions or Inverse Circular Functions. = 96. In Art. 58 we showed that an infinite number of angles, differing by 2, have the same ratios. Accordingly an infinite number of angles will satisfy an equation of the form = sin-1y, 0 = cos-1x, etc. Accordingly for the sake of definiteness we shall (unless otherwise stated) make the following conventions: (1) When we are given either of the equations sin-'y, = tan ly, csc-ly, cot-ly, we shall understand to be an angle, either positive or negative, whose magnitude is not greater than 90°. 1 √2 = sin-1. = (2) When we are given the equations : = cos1y, sec-1y, we shall limit to a positive angle whose magnitude is not greater than 180°. 68 = With these agreements one value and but one will satisfy any of these equations. If, however, is given, we can always write a definite equation. For example, 225° = sin-1 and had known nothing else whatever of ø, by our agreements we would have concluded = − 45°. EXAMPLE 2. Given : = cos-1 8-11-20) By our agreements we know that = 135°. We can now write · ( — — — 2 )· If, on the other hand, we are given = sin-1 97. To express the inverse ratios in terms of a given one. e.g. if o tan-', to express the inverse trigonometric functions in terms of x. = EXERCISE. terms of x. 135° sin-1 But if we had given us sin 0 ; = csc 0; we conclude = 45°. No ambiguity of sign exists. For if x is positive, sin is positive; therefore √1+22 is positive. If x is negative, sin is negative; hence V1+ is positive. Then by convention cos @ is positive; therefore √1+ is positive. If x = cos 0, express the inverse trigonometric functions in Then 98. Let x = sin 0. EXERCISE. Show Put 99. To express two inverse trigonometric functions as a single inverse function; e.g. consider sin-1x + sin-1y. .'. sin-1x = = π 2 .. sin (A + B) = x√1 − y2 + y√1 — x2. A + B = sin-1x + sin-1y = sin-1 (x√1 − y2 + y√1 − x2). EXERCISE. tan-1x X sec-1x ཏྭཱ Express tan-1x + tan-ly as a single inverse trigonometric tan-1 (}); find cos p. sin-1 (-1) find sec; find tan p. 1 0 = tan-1 2 m + 1 x = sin A; .. A sin-1x. = π = cot-1x; 2 y sin B; .. B= sin-1y; = π 2 10. cos ̄1x cos-x. csc-1x. EXAMPLES. XXV. b + a 2. 4 = cos−1 ( − }) find csc p. ; show that (@+4)=— cos-1y = cos-1 (xy ± √(1 − x2) (1 — y2)). tan-11=2 CHAPTER XIII ON THE RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE 100. The three sides and the three angles of any triangle are called its six parts. By the letters A, B, C we shall indicate geometrically, the three angular points of the triangle ABC; algebraically, the three angles at those angular points respectively. A B a By the letters a, b, c, we shall indicate the measures of the sides BC, CA, AB, opposite the angles A, B, C, respectively. 101. I. We know that A + B + C = 180°. (Geom.) 102. Also if A be an angle of a triangle, then A may have any value between 0° and 180°. Hence, 1 (i.) sin ▲ must be positive (and less than 1); (ii) cos ▲ may be positive or negative (but must be numerically less than 1); (iii.) tan A may have any value whatever, positive or negative. 103. Also, if we are given the value of (i.) sin A, there are two angles, each less than 180°, which have the given positive value for their sine. |