Trignometric and hyperbolic equalities: Why the golden ratio?

1. Jul 14, 2011

dimension10

1.

$$\sin \theta = \cos \theta$$

\theta=\frac{\pi}{4}

2.

$$\sin \theta = \tan \theta$$

$$\theta = 0$$

3.

$$\cos \theta = \tan \theta$$

$$\theta =\arcsin (\varphi -1)$$

4.

$$\sin \theta = \csc \theta$$

$$\theta = \frac{\pi}{2}$$

5.

$$\sin \theta =\sec \theta$$

$$\theta$$ does not exist.

6.

$$\sin \theta =\cot \theta$$

$$\theta = \arccos (\varphi -1)$$

7.

$$\cos \theta =\csc \theta$$

$$\theta$$ does not exist.

8.

$$\cos \theta =\sec \theta$$

$$\theta=0$$

9.

$$\cos \theta = \cot \theta$$

$$\theta=\frac{\pi}{2}$$

10.

$$\tan \theta =\csc \theta$$

$$\theta =\arccos(\varphi-1)$$

11.

$$\tan \theta = \sec \theta$$

$$\theta=\frac{\pi}{2}$$

12.

$$\tan \theta = \cot \theta$$

$$\theta=\frac{\pi}{4}$$

13.

$$\csc \theta =\sec \theta$$

$$\theta=\frac{\pi}{4}$$

14.

$$\csc \theta =\cot \theta$$

$$\theta = \arccos (\varphi -1)$$

15.

$$\sec \theta =\cot \theta$$

$$\theta=\arcsin (\varphi - 1)$$

I used quadratic equation for some equalities. Which showed that the golden ration was involved. But my question is "geometrically, why?"

2. Jul 14, 2011

Avijeet

Hi!
Interesting question. I have always been fascinated by the golden ratio which keeps appearing at places where you least expect Will have to think about this one.

3. Jul 15, 2011

dimension10

Oops. There was a LaTeX error.

1.

$$\sin \theta = \cos \theta$$

$$\theta=\frac{\pi}{4}$$

2.

$$\sin \theta = \tan \theta$$

$$\theta = 0$$

3.

$$\cos \theta = \tan \theta$$

$$\theta =\arcsin (\varphi -1)$$

4.

$$\sin \theta = \csc \theta$$

$$\theta = \frac{\pi}{2}$$

5.

$$\sin \theta =\sec \theta$$

$$\theta$$ does not exist.

6.

$$\sin \theta =\cot \theta$$

$$\theta = \arccos (\varphi -1)$$

7.

$$\cos \theta =\csc \theta$$

$$\theta$$ does not exist.

8.

$$\cos \theta =\sec \theta$$

$$\theta=0$$

9.

$$\cos \theta = \cot \theta$$

$$\theta=\frac{\pi}{2}$$

10.

$$\tan \theta =\csc \theta$$

$$\theta =\arccos(\varphi-1)$$

11.

$$\tan \theta = \sec \theta$$

$$\theta=\frac{\pi}{2}$$

12.

$$\tan \theta = \cot \theta$$

$$\theta=\frac{\pi}{4}$$

13.

$$\csc \theta =\sec \theta$$

$$\theta=\frac{\pi}{4}$$

14.

$$\csc \theta =\cot \theta$$

$$\theta = \arccos (\varphi -1)$$

15.

$$\sec \theta =\cot \theta$$

$$\theta=\arcsin (\varphi - 1)$$

I used quadratic equation for some equalities. Which showed that the golden ration was involved. But my question is "geometrically, why?"

4. Jul 15, 2011

pwsnafu

IIRC, $\cos\frac{\pi}{5} = \frac{\phi}{2}$. That's related to the pentagon. Hmm...

5. Jul 15, 2011

dimension10

Here are the hyperbolic equalities.

1.

$$\sinh x = \cosh x$$

$$x= \infty$$

2.

$$\sinh x =\tanh x$$

$$x=0$$

3.

$$\cosh x =\tanh x$$

$${x}_{1}=\frac{- \arcsin (\frac{\sqrt{3}}{2}+\frac{i}{2})}{i}$$

$${x}_{2}=\frac{- \arcsin (\frac{\sqrt{3}}{2}-\frac{i}{2})}{i}$$

4.

$$\sinh x = csch x$$

$$x=arcsinh 1$$

5.

$$\sinh x = sech x$$

$$x=\frac{\ln (2 \varphi +1)}{2}$$

6.

$$\sinh x =\coth x$$

$$x=\frac{\arccos(1-\varphi)}{i}$$

7.

$$\cosh x =csch x$$

$$x=arcsinh \sqrt{\varphi-1}$$

8.

$$\cosh x =sech x$$

$$x=0$$

9.

$$\cosh x =\coth x$$

$$x=arcsinh (\varphi - 1)$$

10.

$$\tanh x = csch x$$

$$x=arccosh (\varphi - 1)$$

11.

$$\tanh x =sech x$$

$$x=arcsinh 1$$

12.

$$\tanh x = \coth x$$

$$x=\infty$$

13.

$$csch x =sech x$$

$$x=arctanh 1$$

14.

$$csch x =\coth x$$

$$x=0$$

15.

$$sech x = \coth x$$

$$x=arcsinh (\varphi -1 )$$

Still a lot of golden ratios.

6. Jul 15, 2011

dimension10

Ok, so thats why its related...

7. Jul 15, 2011

uart

This one doesn't look right. I'm pretty sure there should be a real solution there.

8. Jul 15, 2011

uart

After just doing some calculations I'm pretty sure that the solution to 6. should be

$$x = \pm \, \cosh^{-1} \phi$$

BTW. Inverse hyperbolics can usually be alternatively represented using logs.

Last edited: Jul 15, 2011
9. Jul 15, 2011

dimension10

You mean arccosh right?

10. Jul 15, 2011

dimension10

That works too. But my solution also works. I converted sinh x to - i sin i x and it works.

11. Jul 15, 2011

dimension10

And by the way, what is the LaTeX code for arcsinh, arccosh, sech, cosech?

12. Jul 15, 2011

Mute

Two things:

1) The inverse hyperbolic functions are not "arc(whatever)". They are actually "ar(whatever)". i.e., arsinh(x), arcosh(x), artanh(x), etc.

2) I don't believe there is latex commands for most of these. You typically have to use \mbox{arsinh}(x), etc.

Tests:

$\arsinh(x), \arcosh(x), \artanh(x), \sech(x), \csch(x)$

Check to make sure latex doesn't use the misnamed versions:

$\arcsinh(x), \arccosh(x)$

13. Jul 15, 2011

dimension10

Ok, thanks.

Oh. So it can be written as $$\mbox{arsinh}(x)$$?

14. Jul 15, 2011

Mute

Yep. If you're writing in an actual latex document, you can always define new commands so that you don't always have to use mbox.

For example, writing

\newcommand{\arsinh}{\mbox{arsinh}}

in the top before the document begins would let you use \arsinh as a command.

15. Jul 16, 2011

uart

Yes the "f^{-1}" notation is still very commonly used for both trig and hyp-trig functions. See for example : http://mathworld.wolfram.com/InverseHyperbolicFunctions.html

Yes I know that it works but I was considering being consistent with your original post in which you were clearly only considering real solutions.

For example :
There are no real solutions to that equation, but there definitely are complex solutions. Basically I was pointing out that the expression in question doesn't evaluate to a real number, whereas previously you seemed to be only considering reals.

16. Jul 16, 2011

dimension10

I was just clarifying to make sure it was an inverse hyperbolic function and not something like sin^2...

Are you sure there are complex solutions?

$$\sin \theta =\sec \theta$$

$$\sin \theta =\frac{1}{\cos \theta}$$

$$\sin \theta \cos \theta =1$$

As the maximum of both $$\sin \theta$$ and $$\cos \theta$$ is 1, both $$\sin \theta$$ and $$\cos \theta$$ should be 1. Then as $$\cos \theta=1$$, $$\frac{\theta}{\pi}$$ is a whole number, thus $$\sin \theta=0$$. But we have earlier said that $$\sin\theta=1$$ and $$1 \neq 0$$ so there is no value of theta.

17. Jul 16, 2011

pmsrw3

Last edited by a moderator: Apr 26, 2017
18. Jul 16, 2011

dimension10

Last edited by a moderator: Apr 26, 2017