Formulas

Trig Identities

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