Each angle which has either OP or OP1 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 m • 360° + A, that is, in 2 m. 180° + A, or 2 mπ+a, (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° + 4), 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= tan (n. 180° + A); tan a = tan (nя +a). (4) Since cote = 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 = 1. The least positive angle whose tangent is 1, is Hence 0 Nπ + in which n is any positive or negative whole number. π 4 π Find the general value of 0, and the four least positive values of 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-1 m, tan-1 m, ......., are called inverse trigonometric functions, or anti-trigonometric functions, or inverse circular functions. The symbol "sin-1 m" 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; 1 sin-1m does not denote (sin m)-1, that is, ; sin-1m denotes each sin m 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. For instance, if = in (3), then m =+ if m = 4 = n • 180° + (− 1)" 45°, 1 √2 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 0 or m has a single definite value. For a given value of the sine m, sin-1m or @ has an infinite number of values. The same is the case with each of the other inverse trigonometric functions. Thus the trigonometric functions are singlevalued, and the inverse circular functions are multiple-valued. if tan-11, then = n + = π (Ex. 1, Art. 87), in which n denotes any 4 whole number. The smallest numerical value of an inverse trigonometric function is called the principal value of the inverse function. For instance, the principal value of sin-1 is 30°, of tan-1 (-1) is — 45°, of cos ̄1 (— 1⁄2) is 120°, of sin-1 .... ( is 60°. NOTE 1. In some books the symbols arc sin x, arc cos x, arc tan x, ....., are used for inverse trigonometric functions. These symbols are read, "arc sine x," The derivation of these names is apparent from Art. 79. NOTE 2. Algebraic. If y is a function of x, say f(x), then x also depends on y, and hence, is some function of y. This function of y is called the inverse function of f(x) or y, and is usually denoted by f-1(y). For instance, if y = f(x) = x2, then x = ƒ−1(y) = ±√y. It will be observed in this simple example that, while the function of x has a single value, the inverse function has two values. In other words, y is a single-valued function of x, and x is a two-valued function of y. As shown above, if y = sin x, then x = sin-1y; y is a single-valued function of x, but x is a multiple-valued function of y. It appears from Notes 1, 2, that the English notation for inverse trigonometric functions avoids the old geometrical conceptions of trigonometric functions, and is also more general in character. The inverse trigonometric functions are frequently met in calculus and applied mathematics. 89. Sum and difference of two anti-tangents. verse functions. Exercises on in x= tan-1m, and y = tan-1n. Now tan (x+y) tan x = m, tan y = n. tan +tany (Art. 51) = 1 tan x tan y ; i.e. tan−1m + tan−1 n = tan−1 m +n In a similar manner it can be shown that tan-1 m - tan-1 n = tan-1 m ·n 1 + mn 1 mn (1) (2) EXAMPLES. 1. Find tan-12 ± tan-1. (Compare Ex. 1, Art. 51.) By the tables, taking acute angles only, tan-1 2 = 63° 26′ 4′′.3, tan-1 =18° 26'6", the sum is 81° 52' 10".3, and the difference is 44° 59′ 58′′.3. The slight discrepancy between the results obtained by the two methods is due to the fact that the angles found by the tables are only approximately correct. In the following examples test or verify the result in the manner shown in Ex. 1. 2. Find tan-17± tan-13. 4. Find tan-11 + tan-1}. 3. Find tan-12+ tan-1.5. 5. Find tan-13+ tan-12+ tan-1.6. (SUGGESTION. Find tan-13+tan-12, then combine the result with tan-1.6.) 6. Find 2 tan-11.5, 2 tan-13, 2 tan-12, 3 tan-1.2. 9. Find sin (sin−1 + sin-1) when the angles are between 0° and 90°. 10. When the angles are between 0° and 90°, show that: (a) sin (sin-1 m + sin-1n) = m √1 − n2 ± n√1 — m2. sin-1 m, y = sin-1n.) (b) cos (sin-1m ± sin−1 n) = √1 − n2 √1 — m2 = mn. 11. Find sin (sin-1 + sin-1), cos (sin-1 cos−1}), sin (cos-1 cos-1), sin (tan-14- cos-13), tan (sec-13-sin-1), (a) when the angles are between 0° and 90°, (b) when this restriction is not imposed. 12. Two lines, AB, AC, intersect a horizontal line at B, C, making angles whose tangents are, . Find the angle BAC. 13. Two lines, LM, LN, make angles whose tangents are }, 2, with a horizontal line. Find the angle MLN. 90. Trigonometric equations. Trigonometric equations have appeared in many of the preceding articles. When an angle, say, is the unknown quantity in a trigonometric equation, the complete solution is the general value of ✪ which satisfies the equation. For example, if a be an angle whose sine is s, then the solution of the equation, The given equation shows that sinx and cos x have opposite algebraic signs. Hence, x can only be in the second and fourth quadrants. .. In (a), x = 120°, 300°, etc., its general value is n. 180 - 60°, where n is any positive integer. In (b), x = 150°, 330°, etc.; its general value is n. 180° – 30°, n being any positive integer. 4. Solve the equation sin 50+ sin = sin 3 0. .. 2 sin 30 cos 2 0 = sin 3 0. .. sin 30(2 cos 2 0 − 1) = 0. .. (a) sin 30=0, (b) 2 cos 20 1=0. From (a), 30=0°, 180°, etc.; the general value of 30 is ní (n being any integer). .. 0 = 0°, 60°, etc.; the general value of 30 is пп 3 From (b), cos 2 0 = }. .. 20 = ± 60°, etc.; its general value is 2 nTM ± ૧૧/૧ |