For which value of e is sin e=-1

=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