Question

For which value of e is sin e=-1

Answer 1

Answer:

=ei x

Step-by-step explanation:

Consider the function on the right hand side (RHS)

   f(x) = cos( x ) + i sin( x )

Differentiate this function

   f ' (x) = -sin( x ) + i cos( x) = i f(x)

So, this function has the property that its derivative is i times the original function.

What other type of function has this property?

A function g(x) will have this property if

   dg / dx = i g

This is a differential equation that can be solved with seperation of variables

   (1/g) dg = i dx

   integral (1/g) dg = integral i dx

   ln| g | = i x + C

   | g | = ei x + C = eC ei x

   | g | = C2 ei x

   g = C3 ei x

So we need to determine what value (if any) of the constant C3 makes g(x) = f(x).

If we set x=0 and evaluate f(x) and g(x), we get

   f(x) = cos( 0 ) + i sin( 0 ) = 1

   g(x) = C3 ei 0 = C3

These functions are equal when C3 = 1.

Therefore,

   cos( x ) + i sin( x ) = ei x