### "Relation between Exponential Function & Complex Number" by Ajit Mishra's Online Classroom

### Relationship to exponential function and complex numbers

^{}that the sine and cosine functions are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary:

*e*

^{ ix}, and as above, we can parametrize this circle in terms of cosines and sines, the relationship between the complex exponential and the trigonometric functions becomes more apparent.

Euler's formula can also be used to derive some trigonometric identities, by writing sine and cosine as:

*z*:

*i*

^{ 2}= −1. The sine and cosine defined by this are entire functions. Also, for purely real

*x*,

*sin*,

*cos*) and hyperbolic real (

*sinh*,

*cosh*) counterparts.

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