INTRODUCTION

As functions of a complex variable z, we have seen that both the trigonometric and hyperbolic functions are periodic. Consequently, these functions do not possess inverses that are functions in the strictest interpretation of that word. The inverses of these analytic functions are multiple-valued functions. As we did in Section 17.6, in the examination of the logarithmic function, we shall drop the adjective multiple-valued throughout the discussion that follows.

Inverse Sine

The inverse sine function, written as sin⁻¹z or arcsin z, is defined by

(1)

The inverse sine can be expressed in terms of the logarithmic function. To see this we use (1) and the definition of the sine function:

From the last equation and the quadratic formula, we then obtain

(2)

Note in (2) we did not use the customary symbolism ±, since we know from Section 17.2 that (1 − z²)^1/2 is two-valued. Solving (2) for w then gives

(3)

Proceeding in a similar manner, we find the inverses of the cosine and tangent to be

(4)

(5)

EXAMPLE 1 Values of an Inverse Sine

Find all values of sin⁻¹.

SOLUTION

From (3) we have

With (1 − ()²)^1/2 = (−4)^1/2 = ±2i, the preceding expression becomes

The foregoing result can be simplified a little by noting that log_e( − 2) = log_e(1/( + 2)) = −log_e( + 2). Thus for n = 0, ±1, ±2, . . . ,

(6)
≡

To obtain particular values of, say, sin⁻¹z, we must choose a specific root of 1 − z² and a specific branch of the logarithm. For example, if we choose (1 − ()²)^1/2 = (−4)^1/2 = 2i and the principal branch of the logarithm, then (6) gives the single value

Derivatives

The derivatives of the three inverse trigonometric functions considered above can be found by implicit differentiation. To find the derivative of the inverse sine function w = sin⁻¹z, we begin by differentiating z = sin w:

Using the trigonometric identity cos²w + sin²w = 1 (see Problem 28 in Exercises 17.7) in the form cos w = (1 − sin²w)^1/2 = (1 − z²)^1/2, we obtain

(7)

Similarly, we find that

(8)

(9)

It should be noted that the square roots used in (7) and (8) must be consistent with the square roots used in (3) and (4).

EXAMPLE 2 Evaluating a Derivative

Inverse Hyperbolic Functions

The inverse hyperbolic functions can also be expressed in terms of the logarithm. We summarize these results for the inverse hyperbolic sine, cosine, and tangent along with their derivatives:

(10)

(11)

(12)

(13)

(14)

(15)

EXAMPLE 3 Values of an Inverse Hyperbolic Cosine

17.8 Exercises Answers to selected odd-numbered problems begin on page ANS-46.

In Problems 1–14, find all values of the given quantity.

1. sin⁻¹(−i)
2. sin⁻¹
3. sin⁻¹ 0
4. sin⁻¹
5. cos⁻¹ 2
6. cos⁻¹ 2i
7. cos⁻¹
8. cos⁻¹
9. tan⁻¹ 1
10. tan⁻¹ 3i
11. sinh⁻¹
12. cosh⁻¹ i
13. tanh⁻¹(1 + 2i)
14. tanh⁻¹(−i)