How to write powers of inverse trigonometric functions?

In summary, The use of ##\sin^{-1}\theta## for the inverse function of ##\sin## can be confusing, as it is not the same as ##\arcsin\theta##. It is recommended to use ##\arcsin## and ##\csc## for the functional and multiplicative inverses of ##\sin##, respectively. For multiple iterations of the function, a different notation such as ##\sin^{[n]}## or ##\sin^{\circ n}## may be more suitable."
  • #1
Kuhan
46
0
Does ##(\sin^{-1}\theta)^2 =\sin^{-2}\theta## ?
 
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  • #2
I think we use ##\arcsin## for the functional inverse of ##\sin## and ##\csc## for its multiplicative inverse, instead of ##\sin^{-1}##, in order to avoid this confusion. As ##\sin^n## is multiplicative for ##n>0##, I would say that ##\sin^{-n}=\csc^n##. If one needed to write many times the functional iterates and inverses of ##\sin##, I would recommend to use a notation like ##\sin^{[n]}## or ##\sin^{\circ n}##, which I've found in papers dealing with iterated functions.
 
  • #3
why would you even come across something like that?

would be easier to just use the inverse function mate.
 
  • #4
To put it simply, "[itex]sin^{-1}(x)[/itex]" for the inverse function is an unfortunate notation!
 
  • #5
Alberto7 said:
I think we use ##\arcsin## for the functional inverse of ##\sin## and ##\csc## for its multiplicative inverse, instead of ##\sin^{-1}##, in order to avoid this confusion. As ##\sin^n## is multiplicative for ##n>0##, I would say that ##\sin^{-n}=\csc^n##. If one needed to write many times the functional iterates and inverses of ##\sin##, I would recommend to use a notation like ##\sin^{[n]}## or ##\sin^{\circ n}##, which I've found in papers dealing with iterated functions.

Thanks! now it makes sense. I used to use ##\sin^{-1}\theta## instead of ##\arcsin\theta## . They just aren't the same, I guess. Basically,
##\sin^{-1}\theta=\csc\theta##
 

FAQ: How to write powers of inverse trigonometric functions?

1. How do I write powers of inverse trigonometric functions?

To write powers of inverse trigonometric functions, you will need to use the following formula: (arcsin x)^n = sin^-1(x)^n = (sin^-1 x)^n. This formula can be used for any inverse trigonometric function, such as arcsin, arccos, or arctan.

2. Can I simplify powers of inverse trigonometric functions?

Yes, powers of inverse trigonometric functions can be simplified using basic trigonometric identities. For example, if you have (arcsin x)^2, you can rewrite it as (sin^-1 x)^2 = (sin^-1 x)(sin^-1 x) = sin^-1 (sin x) = x. Keep in mind that simplification may not always be possible depending on the power and specific function.

3. What is the range of powers of inverse trigonometric functions?

The range of powers of inverse trigonometric functions is dependent on the original function being raised to a power. For example, if the original function is arcsin x, which has a range of -π/2 to π/2, then (arcsin x)^2 will have a range of 0 to π/4. It is important to consider the domain and range of the original function when dealing with powers of inverse trigonometric functions.

4. How can I use powers of inverse trigonometric functions in applications?

Powers of inverse trigonometric functions are commonly used in applications involving angles and triangles, such as in navigation or engineering problems. They can also be used to simplify complex trigonometric expressions and solve equations involving inverse trigonometric functions.

5. Are there any special properties of powers of inverse trigonometric functions?

Yes, there are a few special properties of powers of inverse trigonometric functions. One is that the power can be distributed to each term inside the parentheses, similar to how it would be distributed in an algebraic expression. Another property is that the power can be combined with other powers of the same function, using the laws of exponents. Lastly, powers of inverse trigonometric functions can be written as a rational exponent, such as (arcsin x)^1/2.

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