Relationship between hyperbolic cosine and cosine

[mnb96]

Hello,

I am considering the hyperbola $x^2-y^2=1$ and its intersection with the line y=mx. The positive x-coordinate of the intersection is given by: $$x=\sqrt{\frac{1}{1-\tan^2\alpha}}=\sqrt{\frac{\cos^2 \alpha}{\cos(2\alpha)}}=\cos\alpha \sqrt{\sec(2\alpha)}$$ where we used the identity $m=\tan\alpha$.

However, using Euler formulas for cosines does not seem to give the relationship: $\cosh(\alpha)=\cos(i\alpha)$.
Am I using a wrong geometrical definition of hyperbolic cosine? I mean, perhaps the hyperbolic cosine is not simply the x-coordinate of the intersection of a ray with the hyperbola?

 

[tiny-tim]

hello mnb96!

(i'm not quite following your question, but anyway …)

you need to use m = tanhα

 

[HallsofIvy]

Hello,

I am considering the hyperbola $x^2-y^2=1$ and its intersection with the line y=mx. The positive x-coordinate of the intersection is given by: $$x=\sqrt{\frac{1}{1-\tan^2\alpha}}=\sqrt{\frac{\cos^2 \alpha}{\cos(2\alpha)}}=\cos\alpha \sqrt{\sec(2\alpha)}$$ where we used the identity $m=\tan\alpha$.

However, using Euler formulas for cosines does not seem to give the relationship: $\cosh(\alpha)=\cos(i\alpha)$.
Am I using a wrong geometrical definition of hyperbolic cosine? I mean, perhaps the hyperbolic cosine is not simply the x-coordinate of the intersection of a ray with the hyperbola?

Yes, you are. The line y= mx has nothing to do with it. 'cos(t)' is defined as the x-coordinate of the point (x,y) at distance t around the circumference of the circle, $x^2+ y^2= 1$ from (1, 0).

So 'cosh(t)' is the x-coordinate of (x, y) at distance t around the curve $x^2- y^2= 1$ from (1, 0).

 

[mnb96]

thanks for your replies!

@Hallsofivy: when you said "distance around the circumference" you meant distance in terms of arc length of the circumference of the unit circle?

 

[tiny-tim]

@Hallsofivy: when you said "distance around the circumference" you meant distance in terms of arc length of the circumference of the unit circle?

yes he did

arc-distance round a circle is proportional to angle,

and arc-distance round a hyperbola is proportional to hyperangle