### 7.4 Complex Trigonometric Functions: Sine, Cosine, Tangent

From the complex exponential function it is only on step to the complex trigonometric function. In addition to the Euler formula $Math content$ we require the definitions of the hyperbolic functions $Math content$ and $Math content$:

$Math content$

$Math content$

#### 7.4.1 Complex Sine

When shifting the point arrays parallel to the real axis one observes their periodic mapping. The square array is then mapped into a region that is bounded by orthogonal ellipses and hyperbolas. Further details and hints for experiments are given in the description pages of the simulation.

As is to be expected the mapping via the cosine for a phase shift by $Math content$ on the real axis leads to the same result as the mapping for the sine. Fig.7.8 shows this for the same configuration of the $Math content$-plane as in Fig.7.7 with $Math content$. Further details and hints for experiments are given in the description pages of the simulation.

#### 7.4.2 Complex Tangent

In addition to the expected periodicity under shifts parallel to the real axis, the complex tangent $Math content$ shown in Fig.7.8 shows a wealth of interesting phenomena because of its divergence with sign change at odd multiples of $Math content$. Because of the large sensitivity close to the divergences you should use the two number fields to enter exact values for $Math content$ and $Math content$. They can be chosen outside of the intervals covered by the sliders.

Straight lines parallel to the real land imaginary axis here are mapped into closed curves around and through the points $Math content$ and $Math content$. The region with imaginary value large $Math content$ is mapped to the point $Math content$, the region with imaginary values smaller $Math content$ is mapped to the point $Math content$. Further details and hints for experiments are given in the description pages of the simulation.