What's the difference between sin^(-1) and 1/sin?

Question

I have this huge math project due for thursday about everything I learned math-wise so far in my life. I know of sin^(-1) (or asin) and cosec (csc) which is 1/sin. Since x^(-1) = 1/x, is sin^(-1) = 1/sin? If so, it would mean that sin^(-1) and cosec would both be the same thing right? Furthermore, I don't know how to label those functions. I would call them inverse functions but they would both have the same name. So, basically, are sin^(-1) and cosec the same? If not, how would you label them? Thanks, Seb

Answer 1

sin⁻¹ x is most certainly not the same thing as 1/sin x. The superscript -1 has two distinct meanings in mathematics. One of them is the one you mentioned, exponentiation to the power of -1, or equivalently taking the multiplicative inverse. The other, however, is the inverse function -- for instance, if f:[0, ∞) → R is given by f(x) = x², f⁻¹(x) = √x ≠ 1/x² = f(x)⁻¹. In the case of sin⁻¹ x, it means the latter, not the former.

The fact that the same exact notation is used to refer to two different things, often within the same sentence, is frequently confusing. At least in the case of a general function, there is a convention that distinguishes between functional powers and multiplicative powers based on the placement of the exponent -- specifically, if the exponent is placed after the function symbol, but before the parentheses, it refers to functional powers (i.e. f²(x) = f(f(x))), whereas if the exponent is placed after the parentheses, it refers to multiplicative powers (i.e. f(x)² = f(x) * f(x)).

However, due to common usage, this convention is explicitly not followed for trigonometric functions. Rather, an exponent in the same place will mean different things depending on whether it is positive or negative -- when the exponent is positive, it refers to taking the appropriate multiplicative power. So for instance, sin² x = (sin x)² = (sin x) * (sin x). However, when the exponent is -1, it always refers to the inverse function, not the multiplicative inverse. So sin⁻¹ (x) = arcsin x ≠ csc x = 1/sin x = (sin x)⁻¹.

This is an unfortunate convention, because not only does it present a case of operator overloading, where the same notation may refer to two different concepts, but it also violates even the usual conventions with respect to functional powers. It is very much like the infamous "ough" in English, which may have up to ten different pronunciations depending on the word it appears in (see the wikipedia article http://en.wikipedia.org/wiki/Ough ). Well, okay, it's not that bad, but in a field that depends on precise thinking, seeing such a schizophrenic notation is jarring. For that reason, I always use of arcsin x in lieu of sin⁻¹ x in my own writing, so as to avoid any possibility of confusion.