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