XOPA; then XOP1 = 180° - A. Every angle whose terminal line is either OP or OP, has its sine equal to s. Now all angles having OP for a terminal line are obtained by adding all numbers of complete revolutions (positive and negative) to XOP. Hence, these angles are represented by m • 360° + A, i.e. 2 m · 180° + A, (1) in which m is any positive or negative whole number. Similarly, all angles having OP1 for a terminal line are represented by m. 360° + (180° - A), i.e. (2 m + 1) 180° - A. (2) An expression that will include both sets of angles, (1) and (2), will now be obtained. In the expression (1), the coefficient of 180° is even, and the sign of A is positive; in (2), the coefficient of 180° is odd, and the sign of A is negative. Hence, n being any positive or negative whole number, the expression n 180° + (- 1)" A, (3) includes the angles in (1) and (2). This is, accordingly, the general expression for all the angles which have the same sine as A. If radian measure is used, and XOP= α, then (3) takes the form nπ + (−1) α. (4) The result may be thus expressed : sin A=sin {n • 180°+(−1)” A}, sin α=sin {nπ+(−1)"α}. (5) Since cosec 0 1 sin o' the general expression for all angles which have the same cosecant is the same as the general expression for all angles which have the same sine. EXAMPLES. 1. Find an expression to include all angles which have the same sine as 135°. By (3), (4), the expression is n. 180° +(− 1)" 135°, or nã +(−1)n3TM. 4 1 2. Find the general value of the angle whose sine is + least positive angles which have sines equal to + Give the four 1 √2 √2 π n · 180° + (− 1)n 45°, i.e. nã +(− 1)n. To find the four least positive angles, put n = 0, 1, 2, 3, in this expression. 86. General expression for all angles which have the same cosine. Let c be the given value of the cosine. It is required to find an expression to include every angle whose cosine is c. All the angles that have c for a cosine can be represented geometrically, as shown in Figs. 86, 87. In Fig. 86, c is positive; in Fig. 87, c is negative. Let XOP be the least positive angle whose cosine is c, and let XOP= A (in degree measure) = a (in radian_measure). Angle XOP1 =- A=a, also has its cosine equal to c. All angles whose terminal line is OP, have cosines equal to c. All these angles are included in n • 360° + A, i.e. 2 nπ + a, (1) in which n denotes any positive or negative whole number. Also, all angles whose terminal line is OP, have cosines equal to c. All these angles are included in n being as before. Both the expressions, (1), (2), are evidently included in n • 360° ± A, or 2nπ ± α, (3) in which n is any positive or negative whole number. Hence (3) is the general expression for all angles which have the same cosine as A or a. The result may be thus expressed: cos A Since seco = cos (n • 360° ± A); cos a = cos (2 nπ ± a). (4) = 1 cos ' the general expression for all angles which have the same secant is the same as the general expression for all angles which have the same cosine. EXAMPLES. 1. What is the general value of the angles which have the cosine, — ? Give the three least positive angles. The least positive angle whose cosine is, — 1, is 120°. Hence, the general value is, by (3), n·360° ± 120°, i.e. 2 nπ ± }π. On putting n = 0 and 1, the three least positive angles are found to be 120°, 360° - 120°, or 240°, 360+120°, or 480°. These three angles may also be found by means of a figure. 2. Given that cos 0 = least positive values of 0. + √3 find the general value of 0, and find the four 3. As in Ex. 2 when cos 0.99106. 4. As in Ex. 2 when cos 0= .46690. 5. As in Ex. 2 when cos 0.72637. 6. As in Ex. 2 when cos 0.40141. 87. General expression for all angles which have the same tangent. Let t be the given value of the tangent. It is required to find an expression to include all angles which have the same tangent t. All the angles which have the same tangent t can be represented geometrically as in Figs. 88, 89. In Fig. 88, the tangent t is positive, in Fig. 89, it is negative. Let XOPA (in degrees) = a (in radians). Then XOP1 = 180° + A = π + a. Each angle which has either OP or OP, for its terminal line, has its tangent equal to t. All the angles which have OP for a terminal line are included in the expression m360° +4, that is, in 2 m. 180° + A, oг 2 mπ + α, (1) in which m denotes any positive or negative whole number. All the angles which have OP1 for a terminal line are included in the expression m · 360° + (180° + A), that is, in (2 m + 1) 180° + A, or (2 m + 1) π + a. (2) Both these sets of angles, (1) and (2), are included in the expression n. 180° + A, or nπ + α, (3) in which n denotes any positive or negative whole number. Hence (3) is the general expression for all angles which have the same tangent as A or a. The result may be thus expressed: tan A = Since cot 0 = tan (n. 180° + A); tan a = tan (në + a). (4) 1 tan o' the general expression for all angles which have the same cotangent is the same as the general expression for all angles which have the same tangent. EXAMPLES. 1. Find the general value of 0 when tan 0 : π π 4 1. The least positive angle Hence 0 Nπ + in which n is any positive or whose tangent is 1, is negative whole number. Find the general value of 0, and the four least positive values of 0 when : 8. Find the general expression for all angles which have the same sine and cosine. It has been seen that, on 88. Inverse trigonometric functions. the one hand, the value of the sine depends on the value of the angle, and, on the other hand, the value of the angle depends on the value of the sine. If the angle is given, the sine can be determined; if the sine is given, the angle can be expressed. Hence, on the one hand, the sine is a function of the angle, and, on the other hand, the angle is a function of the sine. The latter function is said to be the inverse function of the former. The same holds in the case of each of the other trigonometric functions. Inverse functions are usually denoted by the symbol described below. The two statements: the sine of the angle 0 is m, (1) The symbols sin-1m, cos-1m, tan-1 m, ..., are called inverse trigonometric functions, or anti-trigonometric functions, or inverse circular functions. The symbol "sin-1m" is read, "angle whose sine is m," ," "anti-sine of m," "inverse sine of m," "sine minus one m." It should be carefully remembered that here, 1 is not an algebraical exponent, but is merely part of a mathematical symbol; sin-1m does not denote (sin m)-1, that is, -; sin-1m denotes each sin m 1 - and every angle whose sine is m. The trigonometric functions are pure numbers; the inverse circular functions are angles, and are denoted by the number of degrees or radians in these angles. 1 For instance, if 4 in which n is any whole number. This example illustrates what has already been noted in Arts. 42, 43, 78, namely: For a given value of the angle 0, sin @ or m has a single definite value. For a given value of the sine m, sin-1m or has an infinite number of values. |