Question

2. (a) Solve the complex equation (1+?)?³−[1+?^?(?3)]=0 and list all possible solutions in Euler’s form with principal arguments.
(b) Express the complex number ?=(1−sin?+?cos?)²⁰ in Euler’s form.

Answer 1

(a) We have

Now, let us convert each of the complex number on the right side to Euler's form:-

refers to the point (1,1) which has an argument of 45 degrees and distance from origin is

Hence, we have

Now,

so

Hence,

and

Hence the denominator is Euler's form is

Finally, we can simplify our equation further to get

Thus we have

Now we can use the formula for cube roots of a complex number to find the answer. The formula says that if

then the cube roots are

Hence, from

we have

Therefore the three solutions to the given complex equation are(this is the answer):-

(we convert this last one to principle arguments by subtracting pi from the power)