Comparing the formulas of trigonometric and hyperbolic functions

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?

• Seems fine, probably the fastest way too. – vitamin d Mar 24 at 23:13
• Use the relationship $\cos x=\cosh ix, \sin x=-i\sinh ix$ and then you can convert any trig identity into a hyperbolic one. – Paramanand Singh Mar 25 at 3:15
• 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). – Blue Mar 25 at 4:43

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 $$\sin$$.