# Formulas and Tables

Trigonometry# Relationships between Trigonometric and Hyperbolic Functions

Trigonometric functions: \(\sin x,\) \(\cos x,\) \(\tan x,\) \(\cot x,\) \(\sec x,\) \(\csc x\)

Imaginary unit: \(i\)

- Relationship between the sine and hyperbolic sine

\(\sin {(ix)} = i\sinh x\) - Relationship between the tangent and hyperbolic tangent

\(\tan {(ix)} = i\tanh x\) - Relationship between the cotangent and hyperbolic cotangent

\(\cot {(ix)} = -i\coth x\) - Relationship between the secant and hyperbolic secant

\(\sec {(ix)} = \text{sech }x\) - Relationship between the cosecant and hyperbolic cosecant

\(\csc {(ix)} = -\text{csch }x\) - Sine of a complex number

\(\sin \left( {x + iy} \right) =\) \( \sin x\cosh y \,+\) \( i\cos x\sinh y\) - Cosine of a complex number

\(\cos \left( {x + iy} \right) =\) \( \cos x\cosh y \,-\) \( i\sin x\sinh y\)