Editing Hyperbolic angle (section)

==Imaginary circular angle==
The hyperbolic angle is often presented as if it were an [[imaginary number]], <math display=inline> \cos ix = \cosh x</math> and <math display=inline>\sin ix = i \sinh x,</math> so that the [[hyperbolic function]]s cosh and sinh can be presented through the circular functions. But in the Euclidean plane we might alternately consider circular angle measures to be imaginary and hyperbolic angle measures to be real scalars, <math display=inline> \cosh ix = \cos x</math> and <math display=inline>\sinh ix = i \sin x.</math>

These relationships can be understood in terms of the [[exponential function]], which for a complex argument <math display=inline>z</math> can be broken into [[Even and odd functions|even and odd parts]] <math display=inline>\cosh z = \tfrac12(e^z + e^{-z})</math> and <math display=inline>\sinh z = \tfrac12(e^z - e^{-z}),</math> respectively. Then

<math display=block>e^z = \cosh z + \sinh z = \cos(iz) - i \sin(iz), </math>

or if the argument is separated into real and imaginary parts <math display=inline>z = x + iy,</math> the exponential can be split into the product of scaling <math display=inline>e^{x}</math> and rotation <math display=inline>e^{iy},</math>

<math display=block>e^{x + iy} = e^{x}e^{iy} = (\cosh x + \sinh x)(\cos y + i \sin y).</math>

As [[infinite series]],

<math display=block>\begin{alignat}{3}
e^z      &= \,\,\sum_{k=0}^\infty \frac{z^k}{k!}  &&
          = 1 + z + \tfrac{1}{2}z^2 + \tfrac16z^3 + \tfrac1{24}z^4 + \dots \\
\cosh z  &= \sum_{k \text{ even} } \frac{z^k}{k!} &&
          = 1 + \tfrac{1}{2}z^2 + \tfrac1{24}z^4 + \dots \\
\sinh z  &= \,\sum_{k \text{ odd} } \frac{z^k}{k!} &&
          = z + \tfrac{1}{6}z^3 + \tfrac1{120}z^5 + \dots \\
\cos z   &= \sum_{k \text{ even} } \frac{(iz)^k}{k!} &&
          = 1 - \tfrac{1}{2}z^2 + \tfrac1{24}z^4 - \dots \\
i \sin z &= \,\sum_{k \text{ odd} } \frac{(iz)^k}{k!} &&
          = i\left(z - \tfrac{1}{6}z^3 + \tfrac1{120}z^5 - \dots\right) \\
\end{alignat}</math>

The infinite series for cosine is derived from cosh by turning it into an [[alternating series]], and the series for sine comes from making sinh into an alternating series.