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

However, using Euler formulas for cosines does not seem to give the relationship: [itex]\cosh(\alpha)=\cos(i\alpha)[/itex].

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?

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# Relationship between hyperbolic cosine and cosine

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