Recomposing trig and hyp functions

  • Thread starter Marin
  • Start date
  • #1
193
0
Hi!

I want to recompose the expression of the form:

arcsin(cosx) , arsinh(coshx), arcsin(coshx), arsinh(cosx)

but I can't figure out how this is supposed to work :( I expect square root expressions for the first two and I'm not sure one can recompose the last two without using complex numbers.

Could someone please help me in a way? Thanks in advance :)
 

Answers and Replies

  • #2
uart
Science Advisor
2,788
13
Firstly understand that problems of the form arcsin(cos(x)) are quite different to problems of the form cos(arcsin(x)).

For the first type simply use the "complement" relation that cos(x) = sin(Pi/2-x) to write,

[tex]\sin^{-1}(\cos(x)) = \sin^{-1}(\sin(\frac{\pi}{2}-x)) = \frac{\pi}{2}-x[/tex]

{or 90-x if you prefer to use degrees}.


Since you "were expecting square root type expressions" I assume you where actually thinking about the opposite type of problem such as cos(arcsin(x)).

Here you need to use the relation that [itex]\cos(x) = \sqrt{1-(\sin x)^2}[/itex]

Making this substitution you get,

[tex]\cos(\sin^{-1}(x)) = \sqrt{1-(\sin (\sin^{-1} x))^2}[/tex]
[tex] = \sqrt(1-x^2)[/tex]
 
Last edited:
  • #3
193
0
Thanks, uart!

And what about the hyperbolic functions? There is no Pythagoras for hyperb. functions, is there?

And the mixed type in the field of the real numbers is quite unclear to me :(
 
  • #4
uart
Science Advisor
2,788
13
For hyp-trig functions the equivalent of the pythagorean relation is

[tex]\cosh^2(x) - \sinh^2(x) = 1[/tex]


Yeah for the mixed ones I thinked I'd have to go complex and use cosh(x) = cos(i x) and sinh(x) = -i sin(i x). Maybe someone else can offer a better solution there.
 
Last edited:
  • #5
193
0
So it's:

[tex]\cosh(\arsinhx)=\sqrt{1+\sinh(\arsinhx)}=\sqrt{1+x^2}[/tex]


() is suppose to have \arsinhx inside - what's wrong with the Latex typing it?


And what do we do when we have: arsinh(coshx) hmmmm?
 

Related Threads on Recomposing trig and hyp functions

  • Last Post
Replies
7
Views
4K
Replies
4
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
8
Views
2K
Replies
1
Views
942
Replies
2
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
12
Views
1K
  • Last Post
Replies
6
Views
7K
Top