All these angles may be expressed by n 180° + ( − 1)" 30°, n being any positive or negative integer. Hence

In general, if a is an angle whose sine is u, it may be shown that

sin-1 u = nπ + (− 1)" a.

In a similar manner it may be shown that

79. Principal values. The smallest numerical value of an inverse function is called its principal value, preference being given to positive angles in case of ambiguity.

The principal values of the inverse sine and the inverse cosecant lie between

π

and 2

π

; of the inverse cosine and

the inverse secant, between 0 and π; of the inverse tangent

and the inverse cotangent, between-and

The principal values of an inverse function are sometimes distinguished from the general values by the use of a capital letter.

Thus

Sin-1 =

while

sinn+(-1)"

= •

80. To interpret sin sin-1u and sin-1 sin a.

The expression sin sin-1u is read: the sine of the angle whose sine is u. This sine is evidently u, hence

sin sin-1 u = U.

The expression sin1 sin a is read: the angle whose sine is the sine of a. This angle is evidently a, hence

sin-1 sin α = α,

or more generally,

sin-1 sin α = n +(−1)"α.

Similar relations exist between any direct function and the corresponding inverse function.

Thus

cos cos-1 u = u;

= 2 nπ ± α;

cos-1 cos α = a, or cos-1 cos α =

tan tan-1 u = u ;

tan-1 tan α = a, or tan-1 tan α = n + α, etc.

81. Application of the fundamental relations to angles expressed as inverse functions. The fundamental relations, being true for all angles, must necessarily be true when the angles are expressed as inverse functions.

Thus, letting a tan-1 u in the identity

By expressing the angle of the fundamental relations as an inverse function, we may develop relations between the inverse functions.

82. Given an angle, expressed as an inverse function of u, to find the value of any function of the angle in terms of u.

By the application of one or more of the fundamental

Several illustrations are given below. The method employed can be readily applied to the other functions.

1. To find the value of tan cos ̄1 u in terms of u.

If tan cos-1u is expressed in terms of the cosine of cos-1 u, the problem is solved, since cos cos-1 u = u.

This result may be obtained geometrically. Since u is given, it is evident that cos-1u represents, among others, two positive angles, a1 and α, each less than 360°.

Let us assume u positive and let us construct these angies defined by cos ̄1 u.

Then from the figure and the definition of the tangent,

represents all angles coterminal with either a1 or α2, we have

2. To find the value of sec cot-1 u in terms of u.

To solve this problem it is only necessary to express sec cot-1u in terms of cot cot-1 u.

This result may be obtained geometrically. Construct the angles given by cot-1u. Let us assume in this problem that u is negative and hence that

u is positive. If the

1

B

cotangent of an angle is negative the angle must terminate in either the second or fourth

quadrant. Since OA and OB are

the terminal lines of a1 and a respectively, and since the terminal line is always positive, we have

or by considerations similar to those in the previous ex

3. To find the value of sin cos ̄1 u in terms of u.

We have sin cos-1 u = ±√1-(cos cos ̄1 u)2 = ± √1 — u2.

4. To find the value of cot vers-1u in terms of u.

83. Some inverse functions expressed in terms of other inverse functions.

1. To express cos-1 u in terms of an inverse tangent.

u2

From tan cos-1 u =

, (Art. 82, prob. 1) by

и

taking the inverse tangent of each member (Art. 80), there results

cos-1 u = tan-1 ± √1 — u2

и

2. To express the cot-1 u in terms of an inverse secant.

3. Similarly from sin cos1 u±√1-u2 there results cos1 u = sin1 (± √1 — u2).

By the method exemplified in Arts. 82 and 83 it is possible to express any inverse function in terms of any other inverse function.

In applying the above formulas care must be exercised in selecting the angles, since each inverse function represents an infinite number of angles and one member of the equation may represent angles not represented by the other. For example, in problem 1, if u be positive the cosu represents angles terminating in the first and fourth quadrants; but tan-1 ± V1 — u2

represents angles terminat

ing in the second and third quadrants as well as angles terminating in the first and fourth quadrants.