20 Replies Latest reply: Sep 3, 2006 12:00 AM by ptc-1368288

# Inverse Sine Function

Using Rads, the calculus book gives the inverse sign function, sin^-1 0 = 0 as part of a proof. In Mathcad this would be sin(0)^-1 = 0 or alternately, (1/(sin(0)) = 0.

If I input this into my scientific calculator using the asin function it shows the formula as, sin^-1 0 and gives the result, 0 as in the book, but when I key the equation into Mathcad I get an error. I.e., division by zero.

What am I missing?

Also, is there a way to show actual Mathcad examples in this forum?

Joseph A. Rybczyk
• ###### Inverse Sine Function
"In Mathcad this would be sin(0)^-1 = 0 or alternately, (1/(sin(0)) = 0.

____________________

The inverse of sin(x) is asin(x) [for arcsin(x)].

Search for "Inverse Function" in this collab (maybe InverseFunction ?) there is wealth of work sheets (11.2a) ready to use .

jmG
• ###### Inverse Sine Function
On 8/22/2006 11:47:50 AM, Rybczyk wrote:
>gives the inverse sign

You mean the inverse sine function.

>function, sin^-1 0 = 0 as part
>of a proof. In Mathcad this
>would be sin(0)^-1 = 0 or
>alternately, (1/(sin(0)) = 0.

No, it wouldn't. sin^-1(x) is a notation for the inverse sine of x. That's not the same thing as sin(x)^-1, which is the same as 1/sin(x). In Mathcad, type asin(x). You can also get the notation you are referring to, although it's more trouble. See the attached worksheet.

>If I input this into my
>scientific calculator using
>the asin function it shows the
>formula as, sin^-1 0 and gives
>the result, 0 as in the book,
>but when I key the equation
>into Mathcad I get an error.
>I.e., division by zero.
>
>What am I missing?

Sin(0)=0. Therefore 1/sin(0) gives you an error.

>Also, is there a way to show
>this forum?

Yes. You can attach a Mathcad file. When you are posting a message, there is a check box at the top for "Attach file".

Richard
• ###### Inverse Sine Function
Richard,

It seems it is not quite as simple as it first appeared to be. Take a look at the attachment. I apparently have stumbled on an area of Mathcad that can stand improvement.

I don't intend to pursue this matter any further on the Mathcad forum, but I will use great care when dealing with such convention issues in the future.

Joseph A. Rybczyk
• ###### Inverse Sine Function
Richard, you have answered my question. Of all the years I've been using Mathcad, I was unaware of what you showed in the attachment involving the prefix operator. As it turns out, there are two different ways of inputting the expression in Mathcad, of which only the one using the prefix operator is correct. Thank you very much.

I will take time to look at the other responses as soon as I can, and wish to thank everyone for their inputs.

As far as I'm concerned, my question has been answered. Now I have to do some research to pull it all together in my head.

Thanks again.

Joseph A. Rybczyk
• ###### Inverse Function
What's your point? The fact that some functions are multivalued has nothing to do with whether the notation asin and arcsin mean the same thing.

Richard
• ###### Inverse Function
The sine function is not injective, hence it does not have an inverse function. The inverse of the sine function is a relation, sometimes called a multivalued function (although if you look at the definition of a function that is an oxymoron). The arc sine function, notated in Mathcad as asin, is the principle value of the inverse sine and is a function. But it is not the inverse function to the sine function (which does not exist), as it satisfies only one of the two conditions for an inverse function. As you have domonstrated in your picture.

� � � � Tom Gutman
• ###### Inverse Sine Function
If you want to split hairs, from Mathworld:

"The inverse sine is the multivalued function sin^(-1)z (Zwillinger 1995, p. 465), also denoted arcsin(z) (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 307; Jeffrey 2000, p. 124), that is the inverse function of the sine. The variants Arcsin(z) (e.g., Bronshtein and Semendyayev, 1997, p. 69) and Sin^(-1)z are sometimes used to refer to explicit principal values of the inverse sine, although this distinction is not always made (e.g,. Zwillinger 1995, p. 466). Worse yet, the notation arcsin(z) is sometimes used for the principal value, with Arcsin(z) being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). Note that in the notation sin^(-1)z (commonly used in North America and in pocket calculators worldwide), sin(z) is the sine and the superscript -1 denotes the inverse function, not the multiplicative inverse."

asin is just a variant of arcsin used in some programming languaes, including Mathcad.

Richard
• ###### Inverse Sine Function
Richard,

The matter is not splitting hairs.

My interpretation was like Valery,
i.e: InverseFunction for "inversing a Function"

What the collab wanted ?

1. InverseFunction ?
2. f(x)^-1 ?

I bet most collab had interpreted "InverseFunction"

f(x)^-1 is rather trivial and primitive for a collab with years of Mathcad .
That he wanted a user definition is another story.

jmG

• ###### Inverse Sine Function
On 8/23/2006 10:06:39 AM, jmG wrote:
>Richard,
>
>The matter is not splitting
>hairs.

In this context it is. Joseph just wanted to know why sin(x)^-1 does not give the same result as sin^-1(x). The answer to that is just a question of math notation. The fact that the inverse function can be considered to be multivalued or only valid over it's principle values is irrelevant to the question. Drawing a distinction between the notaions asin and arcsin is certainly splitting hairs. Especially when the distinction is not correct.

Richard
• ###### Inverse Sine Function
"Richard, you have answered my question"
____________________

... as a "user definition" = YES
... as the InverseFunction = NO

and that kind of user definition, you should reject without pulling anything off your head . Have you done some search in this collab ?

jmG
• ###### Inverse Sine Function
First, I really do want to thank everyone for their contributions. Basically, however, Richard is right about my objective. I simply needed to know how to properly input a standard mathematical operation into Mathcad. I am not too interested in how the original definitions came about. I assume them to be correct if they are given in standard textbooks.

Ideally, we should be able to input math content into a computer math program exactly as it appears in the textbook. I.e., the same way we would write it down if we were going to do it manually using pencil and paper. When we start playing around with the convention as to how it should be written, everything becomes too confusing.

As I acknowledged previously, I was unaware of the need to invoke the prefix operator as Richard showed. I was also somewhat confused as to why Mathcad, in the process of evaluation, sometimes returned an arc result, e.g., asin, etc. when I inputted inverse content in one form or another. In other words, what is the relationship between arc functions and inverse functions. With the insight I got from Richard's input, I was able to clarify everything to the point that I am no longer confused by it all and can get back to what I was originally doing when the problem came up.

See the attachment for my final conclusions.

Joseph A. Rybczyk
• ###### Inverse Sine Function
To all,

After all of this, I found the following at the end of the book. (See attachment)

To jmG,

What is the procedure you use to post formulas directly to the board?

Joseph A. Rybczyk
• ###### Inverse Sine Function
On 8/23/2006 10:34:20 AM, Rybczyk wrote:
>Richard,
>
>It seems it is not quite as simple as it first appeared to be. Take a look at the attachment. I apparently have stumbled on an area of Mathcad that can stand improvement.

I don't intend to pursue this matter any further on the Mathcad forum, but I will use great care when dealing with such convention issues in the
future.
>
>Joseph A. Rybczyk
___________________

1. respect conventions = godd idea.
2. follow trig rules = is better idea (in your red case)
3. pursue the issue = up to you (often we help)

I have a full sheets of those things, but hesitate to pass as too many use tricks to make formulas in papers not reproducible by non "Mathcader chevron"

jmg

• ###### Inverse Sine Function
Reciprocal and Inverse are not synonymous,
especially in math context and in trig context.

jmG
• ###### Inverse Sine Function
True, the prefix operator form does not work for all trig functions. Quite annoying. This has been commented on before, and Mathsoft is aware of it. Hopefully it will get fixed at some point in te future.

As far as sin^n(x) goes, n=-1 is a special case. sin^2(x) means the same thing as sin(x)^2. It is an unfortunate quirk of math notation that sin^-1(x) is not the same as sin(x)^-1.

Richard
• ###### Inverse Function
"The sine function is not injective, hence it does not have an inverse function."
______________________

Can't be more right !

See Abramowitz:
"Inverse Circular Functions"
Definitions
4.4.1
4.4.2
4.4.3

jmG
• ###### Inverse Sine Function
On 9/3/2006 11:21:01 AM, Rybczyk wrote:
>To all,
>
>After all of this, I found the
>following at the end of the
>book. (See attachment)
>
>
>To jmG,
>
>What is the procedure you use
>to post formulas directly to
>the board?
>
>Joseph A. Rybczyk
_______________________________

The procedure is few seconds work:

1. capture the equation
2. reduce colors
3. store in *.gif file
4. attach the file

To have it displayed, it follows a universal set of commands, on a new line type

that will display the corresponding image

but you need two more keystrokes

< in the front of img src ....
and a closing > after terminating .gif

This type of command will attach *.bmp, *.jpg, *.tif, *.png ... ?

asin is the inverse of sin in the sense of reflection but asin has its own set of approximation found in Abramowitz "Inverse Circular functions"
What you are doing is 1/sin(x) and asin(x) ,i.e: two different animals .

Visit Wolfram.

jmG

...