Sine of the times.

[Matt Agajanian]

Hi all.

Just befuddled. How did sin(x)² (cosh(x)², csc(x)², etc.) come to be interpreted as sin² x or (sin x)², etc.? To me sin(x)² represents x x² SIN. But, clearly, I must have missed the memo.

[KeithB]

Because computers and calculators have polluted your thinking. You are using the computer function notation. In math texts there are no parens. Attached is a page from Big Red showing the entry for sin.
   

Note that it is just sin z. So putting sin^2 z removes any ambiguity between
sin z^2 or (sin z)^2.

[robve]

(06-12-2024 07:32 PM)Matt Agajanian Wrote:  Just befuddled. How did sin(x)² (cosh(x)², csc(x)², etc.) come to be interpreted as sin² x or (sin x)², etc.? To me sin(x)² represents x x² SIN. But, clearly, I must have missed the memo.

When I first saw this back in middle or high school it confused me too. I thought sin²x might stand for sin(sin(x)) as in function composition. After all, x² = xx hence f²x=ffx so why not? It's my comp sci slash physics brain I suppose.

- Rob

[Nigel (UK)]

(06-12-2024 07:32 PM)Matt Agajanian Wrote:  Hi all.

Just befuddled. How did sin(x)² (cosh(x)², csc(x)², etc.) come to be interpreted as sin² x or (sin x)², etc.? To me sin(x)² represents x x² SIN. But, clearly, I must have missed the memo.

The problem with \(\sin(x)^2\) is that it is ambiguous - is it just \(x\) that's being squared, or is it \(\sin(x)\)? Writing \(\sin^2x\) makes the intent plain without the need for (extra) brackets. Since squared trig functions are so common, it's a very popular time-saving notation.

You did indeed miss the memo. I've found a reference (scroll down the page to find it) to Laplace having used this notation; Gauss found this odious, because he felt that \(\sin^2(x)\) should mean \(\sin(\sin(x))\) (and he has a point!) but since the repeated sine is used so rarely this isn't an issue in practice.

Nigel (UK)

(Added) Just to confuse things, \(\sin^{-1}x\) means the inverse sine of \(x\), not \(1/\sin(x)\)!

[Matt Agajanian]

(06-12-2024 08:58 PM)Nigel (UK) Wrote:  

(06-12-2024 07:32 PM)Matt Agajanian Wrote:  Hi all.

Just befuddled. How did sin(x)² (cosh(x)², csc(x)², etc.) come to be interpreted as sin² x or (sin x)², etc.? To me sin(x)² represents x x² SIN. But, clearly, I must have missed the memo.

The problem with \(\sin(x)^2\) is that it is ambiguous - is it just \(x\) that's being squared, or is it \(\sin(x)\)? Writing \(\sin^2x\) makes the intent plain without the need for (extra) brackets. Since squared trig functions are so common, it's a very popular time-saving notation.

You did indeed miss the memo. I've found a reference (scroll down the page to find it) to Laplace having used this notation; Gauss found this odious, because he felt that \(\sin^2(x)\) should mean \(\sin(\sin(x))\) (and he has a point!) but since the repeated sine is used so rarely this isn't an issue in practice.

(Added) Just to confuse things, \(\sin^{-1}x\) means the inverse sine of \(x\), not \(1/\sin(x)\)!

You see my point exactly! At 58, an avid calculator user and collector since six years of age, I've grown up with sin⁻⁻¹(x) & sin⁻¹ x notation. So, that's first nature to me.

What I've read from a Reddit response to post I've made of this same question, I was told that since EOS, WriteView, and DAL do not allow the entry of sin² notation and immediately enter sin(, you cannot rewrite it as sin² but sin(x)² is the available option and entry method, kids nowadays have been indoctrinated that sin(x)² is now the written and textbook standard (!blasphemy! in my book).

[carey]

(06-12-2024 10:51 PM)Matt Agajanian Wrote:  What I've read from a Reddit response to post I've made of this same question, I was told that since EOS, WriteView, and DAL do not allow the entry of sin² notation and immediately enter sin(, you cannot rewrite it as sin² but sin(x)² is the available option and entry method, kids nowadays have been indoctrinated that sin(x)² is now the written and textbook standard (!blasphemy! in my book).

The HP 17bii avoided this issue by not including any trigonometric functions! :)

[Albert Chan]

I like sin(x)^2 notation. Say out loud, it is the same order: sine x squared.

Inverse Trig Notation: What Do sin^-1 and arcsin Mean?

Quote:you find that early forms of "arcsin" arose in the 1700s, and the sin^2 notation came at the same time;
sin^-1 was introduced in 1813.

This may explain why sin^2(x) cannot mean sin(sin(x)), I wish it did.
Notation were already taken long time ago.

[Matt Agajanian]

(06-13-2024 05:27 PM)Albert Chan Wrote:  I like sin(x)^2 notation. Say out loud, it is the same order: sine x squared.

This may explain why sin^2(x) cannot mean sin(sin(x)), I wish it did.
Notation were already taken long time ago.

Although, the statement itself, without punctuation as you have it can mean two things:

Sine of x squared: (sin(x))²
Sine of x squared: sin(x²)

But Sine of the square of x describes the order: first take the square of x. Then take the sine of that value,

Whereas The square of the sine of x means to take the sine of x then square that value.

[Albert Chan]

Ah, punctuation! Thanks!

sin(x)^2 == sine x, squared
sin(x^2) == sin of squared x

[Matt Agajanian]

Just a note. My above comment wasn’t meant for some show of intellectual superiority. I’m simply pointing out that even the English language needs some means to ensure clarity and precision of meaning.

Full apologies to anyone offended by my grammar/language police efforts.

[WillMarchant]

Please pardon my ignorance, but what’s “Big Red”?

(06-12-2024 08:54 PM)KeithB Wrote:  Attached is a page from Big Red showing the entry for sin.

[Maximilian Hohmann]

Hello,

(06-14-2024 10:34 AM)WillMarchant Wrote:  Please pardon my ignorance, but what’s “Big Red”?

From the contents it must be this book (the clipping above is on page 71): https://personal.math.ubc.ca/~cbm/aands/...stegun.pdf

Although mine is blue and the one in the link is not red either.

Regards
Max

[KeithB]

Wow, I thought this was a common nickname. That is what my professor in college called it.
Yes, Abramowitz and Stegun, Handbook of Mathematical functions.

   

In the early eighties, I went down to the Government printing office store at a mall in downtown LA to pick up a copy. You used to be able to buy Topo maps there, too.

[Matt Agajanian]

(06-14-2024 10:48 AM)Maximilian Hohmann Wrote:  Hello,

(06-14-2024 10:34 AM)WillMarchant Wrote:  Please pardon my ignorance, but what’s “Big Red”?

From the contents it must be this book (the clipping above is on page 71): https://personal.math.ubc.ca/~cbm/aands/...stegun.pdf

Although mine is blue and the one in the link is not red either.

Regards
Max

WOWZA! Thanks! Now I’ve got a copy of this landmark encyclopedia!!

[rprosperi]

(06-15-2024 01:10 AM)Matt Agajanian Wrote:  

(06-14-2024 10:48 AM)Maximilian Hohmann Wrote:  Hello,


From the contents it must be this book (the clipping above is on page 71): https://personal.math.ubc.ca/~cbm/aands/...stegun.pdf

Although mine is blue and the one in the link is not red either.

Regards
Max

WOWZA! Thanks! Now I’ve got a copy of this landmark encyclopedia!!

Printed copies of this historically and technically amazing book are surprisingly common and readily and cheaply available; for example, on ABE there's a copy there now for $10.00 with free shipping.

[johnb]

(06-12-2024 08:58 PM)Nigel (UK) Wrote:  Gauss found this odious, because he felt that \(\sin^2(x)\) should mean \(\sin(\sin(x))\) (and he has a point!) but since the repeated sine is used so rarely this isn't an issue in practice.

(Added) Just to confuse things, \(\sin^{-1}x\) means the inverse sine of \(x\), not \(1/\sin(x)\)!

Mathematician Kenneth Iverson also balked at these inconsistencies in math notation, and sought to fix it once and for all.

APL was the result.


"Be careful what you wish for: you may get it..."

[carey]

(06-18-2024 07:49 PM)johnb Wrote:  Mathematician Kenneth Iverson also balked at these inconsistencies in math notation, and sought to fix it once and for all.

APL was the result.

"Be careful what you wish for: you may get it..."

For anyone interested in array languages, of which APL is the most notable example, there's a podcast on array languages "The Array Cast". A recent episode was an interview with the developer of a free APL system (NARS2000) that includes a toolbar of APL glyphs (no APL keyboard needed), so you can decide if APL was a bad idea, a good idea, or just fun.

[bxparks]

(06-15-2024 01:18 AM)rprosperi Wrote:  Printed copies of this historically and technically amazing book are surprisingly common and readily and cheaply available; for example, on ABE there's a copy there now for $10.00 with free shipping.

Yup, Abramowitz and Stegun was the bible for many years in school. I've never heard it called the "Big Red" either. Everyone around me had the Dover edition which is a big blue book. I have the 9th Dover printing, which claims to be same as the 10th printing (December 1972) of the original, with some additional corrections. As far as I can tell from Amazon, the 9th Dover printing is still the most recent. The cheapest used price at Amazon is $7.91, incl shipping.

[KeithB]

Mine was actually printed by the US Gov printing office.

[Felix Gross]

The current seminal work, not least for notational questions seems to be

https://dlmf.nist.gov

NIST Handbook of Mathematical Functions

This is intended to superseed Abramowitz, Stegun. The tables are gone, but the equations at still there, lots of them. The webpage is updated constantly.

The printed version is a big red book.

Felix