3
$\begingroup$

I just tried to memorize hyperbolic formulas, and realized that there are lots of similarity between formulas for trigonometric and hyperbolic functions. for example: $$\cos^2x+\sin^2x=1\quad\quad\quad\quad\quad\quad\quad1+\tan^2x=\sec^2x$$ $$\cosh^2x-\sinh^2x=1\quad\quad\quad\quad\quad\quad\quad1-\tanh^2x=\text{sech}^2 x$$ And so on. When I look at it closer I see they are almost the same formulas but the sign comes before a $\sinh$ or $\tanh$ are different than the sign of $\sin$ ,$\tan$ in trig functions. To justify that, I considered the fact that parametric points $(\cos\theta,\sin\theta)$ placed on the circle $x^2+y^2=1$ and points $(\cosh \theta,\sinh \theta)$ are placed on hyperbola $x^2-y^2=1$, so we see the different signs comes before $y^2$ and because $\sin$ and $\sinh$ represents the vertical distance, these signs are different in all formulas.

Is my justification right?

Is there better way to justify this different in the formulas?

$\endgroup$
3
  • 4
    $\begingroup$ Seems fine, probably the fastest way too. $\endgroup$ – vitamin d Mar 24 at 23:13
  • 1
    $\begingroup$ Use the relationship $\cos x=\cosh ix, \sin x=-i\sinh ix$ and then you can convert any trig identity into a hyperbolic one. $\endgroup$ – Paramanand Singh Mar 25 at 3:15
  • $\begingroup$ You might consider Osborn's Rule [of Thumb] (see this answer). For a visual mnemonic, see the edit to this answer, which depicts the hyperbolic trig functions as segments related to a hyperbola; it's a counterpart to what I call the "Fundamental Trigonograph" for circular trig functions (as mentioned, eg, in this answer, which provides a link to a note with an in-depth discussion). $\endgroup$ – Blue Mar 25 at 4:43
2
$\begingroup$

It is probably better to think of the difference in algebraic sign as what distinguishes trigonometry from hyperbolic trigonometry. Consider what happens when $\theta$ is close to zero. What happens to $\sin\theta$ and $\sinh\theta$? Same question for $\cos\theta$ and $\cosh\theta$. Those are the properties that you should consider the most basic.

If you understand complex numbers and the natural logarithm, it will be helpful to look at Euler's formula https://brilliant.org/wiki/hyperbolic-trigonometric-functions/ Notice the expressions for $\cosh$ and $\sinh$ are simpler than those for $\cos$ and $\sin$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.