Roots, signs and abs

  • Thread starter Jhenrique
  • Start date
  • #1
685
4
By pythagorean identity, ##\sin(x)^2 + \cos(x)^2 = 1##, so ##\sin(x) = \sqrt{1 - \cos(x)^2}##; also, ##\sinh(x)^2 - \cosh(x)^2 = - 1##, therefore ##\sinh(x) = \sqrt{\cosh(x)^2 - 1}##.

Happens that the last equation is incorrect, here is a full list of the correct forms for the hyperbolics:
https://de.wikipedia.org/wiki/Hyperbelfunktion#Umrechnungstabelle and here is a full trigonometric list for comparation: https://es.wikipedia.org/wiki/Identidades_trigonométricas#Relaciones_b.C3.A1sicas.

So, why the 'normal' trigonometrics no needs of completary functions, like Abs and Sgn, and the hyperbolic trigonometrics needs in some case?
 

Answers and Replies

  • #2
575
76
From the Wiki that you linked ... immediately above the table that is apparently in question:

De estas dos identidades, se puede extrapolar la siguiente tabla. Sin embargo, nótese que estas ecuaciones de conversión pueden devolver el signo incorrecto (+ ó −).
 
  • #3
34,508
6,193
By pythagorean identity, ##\sin(x)^2 + \cos(x)^2 = 1##, so ##\sin(x) = \sqrt{1 - \cos(x)^2}##;
No. You omitted the ##\pm##.
##\sin(x) = \pm \sqrt{1 - \cos(x)^2}##
also, ##\sinh(x)^2 - \cosh(x)^2 = - 1##, therefore ##\sinh(x) = \sqrt{\cosh(x)^2 - 1}##.
Again, no, same problem as above.
##\sinh(x) = \pm \sqrt{\cosh(x)^2 - 1}##
https://de.wikipedia.org/wiki/Hyperbelfunktion#Umrechnungstabelle[/url] and here is a full trigonometric list for comparation: https://es.wikipedia.org/wiki/Identidades_trigonométricas#Relaciones_b.C3.A1sicas.

So, why the 'normal' trigonometrics no needs of completary functions, like Abs and Sgn, and the hyperbolic trigonometrics needs in some case?
 
  • #4
685
4
Yeah, I like of omit +/- because, by definition, a root square have 2 roots...
 
  • #5
34,508
6,193
Yeah, I like of omit +/- because, by definition, a root square have 2 roots...
No, that's not the definition. The square root of a positive real number has one value, not two.


It's true that real numbers have two square roots -- one positive and one negative -- but the expression ##\sqrt{x}## represents the principal square root of x, a positive real number that when multiplied by itself yields x.

If a square root represented two values, there would be no need to write ##\pm## in the quadratic formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

When you start with sin2(x) + cos2(x) = 1 and solve for sin(x), you need ##\pm## in there, otherwise you are getting only the positive value.
 
  • #6
685
4
And if you have ##x=y^6## ? You'll write ##\pm\sqrt{\pm\sqrt{\pm\sqrt{x}}}## ? Not is better let that the ##\sqrt[n]{x}## represents the n roots?
 
  • #7
pwsnafu
Science Advisor
1,080
85
And if you have ##x=y^6## ? You'll write ##\pm\sqrt{\pm\sqrt{\pm\sqrt{x}}}## ? Not is better let that the ##\sqrt[n]{x}## represents the n roots?
On the reals there are only two roots: ##\sqrt[6]{x}## and ##-\sqrt[6]{x}##.
Jhenrique, you have reals and complex numbers mixed up.
 
  • #8
Mentallic
Homework Helper
3,798
94
And if you have ##x=y^6## ? You'll write ##\pm\sqrt{\pm\sqrt{\pm\sqrt{x}}}## ? Not is better let that the ##\sqrt[n]{x}## represents the n roots?
If you take the square root of both sides of

[tex]y^6=x[/tex]

you get

[tex]y^3=\pm\sqrt{x}[/tex]
 
  • #9
34,508
6,193
And if you have ##x=y^6## ? You'll write ##\pm\sqrt{\pm\sqrt{\pm\sqrt{x}}}## ? Not is better let that the ##\sqrt[n]{x}## represents the n roots?
Let's make it simple.
##y^6 = 64##
##\Rightarrow y = \pm \sqrt[6]{64} = \pm 2##

As it turns out, there are four other sixth roots of 64, but they are all complex. The only real sixth roots of 64 are 2 and -2.
 

Related Threads on Roots, signs and abs

  • Last Post
2
Replies
30
Views
3K
Replies
8
Views
3K
  • Last Post
Replies
9
Views
6K
Replies
10
Views
2K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
17
Views
915
  • Last Post
Replies
23
Views
3K
  • Last Post
Replies
7
Views
3K
Replies
3
Views
2K
Top