Formulas and Tables

Trigonometry

Relationships between Trigonometric and Hyperbolic Functions

Angles (arguments of functions): \(x,\) \(y\)
Trigonometric functions: \(\sin x,\) \(\cos x,\) \(\tan x,\) \(\cot x,\) \(\sec x,\) \(\csc x\)

Hyperbolic functions: \(\sinh x,\) \(\cosh x,\) \(\tanh x,\) \(\coth x,\) \(\text{sech }x,\) \(\text{csch }x\)
Imaginary unit: \(i\)

  1. Relationship between the sine and hyperbolic sine
    \(\sin {(ix)} = i\sinh x\)
  2. Relationship between the tangent and hyperbolic tangent
    \(\tan {(ix)} = i\tanh x\)
  3. Relationship between the cotangent and hyperbolic cotangent
    \(\cot {(ix)} = -i\coth x\)
  4. Relationship between the secant and hyperbolic secant
    \(\sec {(ix)} = \text{sech }x\)
  5. Relationship between the cosecant and hyperbolic cosecant
    \(\csc {(ix)} = -\text{csch }x\)
  6. Sine of a complex number
    \(\sin \left( {x + iy} \right) =\) \( \sin x\cosh y \,+\) \( i\cos x\sinh y\)
  7. Cosine of a complex number
    \(\cos \left( {x + iy} \right) =\) \( \cos x\cosh y \,-\) \( i\sin x\sinh y\)