Quiz+1+Solutions

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 * QUIZ 1 SOLUTIONS**


 * 1) a) Express the complex number c=(1+//i//)^3 in both Cartesian and polar forms.**

__Step 1: Expand c= (1+//i//)^3:__ c= (1+//i//)(1+2//i//+//i//^2) c= (1+//i//)(1+//i//-1) c= (1+//i//)(2//i//) c= 2//i//+2//i//^2 c= 2//i//-2 c= -2+2//i//

__Step 2: Rewrite in Cartesian form:__ c= (-2,2), in the complex plane

__Step 3: Rewrite in polar form:__ We need to rewrite c in the following form: , where the complex number z in this case is c

(i) Find θ by solving θ= arctan(y/x). Since, as shown in Step 2, y= 2 and x= -2, (y/x)= (2/-2)= -1. Thus, we have θ= arctan(-1)= 3π/4 for 0≤θ≤2π (ii) Find r by solving Since x=2 and y=-2, we have: r=sqrt{(-2)^2+ (2)^2} r=sqrt {4 + 4} r=sqrt {8} (You can simplify this further, but you do not need to)

(iii) Bring it all together: So, we have c= sqrt{8}(e^(//i//3π/4)) in polar form


 * 1) (b) Using polar coordinates, find all the cube roots of c. (Hint: A cube root is a complex number such that z^3 = c. How many complex solutions does this equation have?)**

If we rewrite z^3= c as z^3 - c = 0, we see that the equation will have exactly three solution (by the __Fundamental Theorem of Algebra__). If we write c is polar form (which we solved in part (a)), we have c= sqrt{8}(e^(//i//3π/4)). So, z^3=sqrt{8}(e^(//i//3π/4))(e^(//i//2πk) for any integer k.

Now, solve for z by raising both sides to the (1/3)rd power: z^3 = sqrt{8}(e^(//i//3π/4))(e^(//i//2πk)) z^3(1/3) = sqrt{8}(1/3)(e^(//i//3π/4)(1/3))(e^(//i//2πk)(1/3)) z = sqrt{2}(e^(//i//π/4)(e^(//i//2πk/3))

To find the three solutions, we first restrict our θ so that 0≤θ≤2π (to avoid repeats), and solve for when k=0,1, and 2 (we stop at two since, after 2, θ becomes greater than 2π and, therefore, cycles).

__//Solve for k=0//__ z//1//= sqrt{2}(e^(//i//π/4)(e^(//0))// z//1// = sqrt{2}(e^(//i//π/4)(1) z//1// = sqrt{2}(e^(//i//π/4)

__//Solve for k=1//__ //z2// = sqrt{2}(e^(//i//π/4))(e^(//i//2π/3)) //z2// = sqrt{2}(e^(//i//11π/12)

__//Solve for k=3//__ //z3// = sqrt{2}(e^(//i//π/4))(e^(//i//4π/3)) //z3// = sqrt{2}(e^(//i//19π/12)

The three solutions to z^3= c are z//1// = sqrt{2}(e^(//i//π/4), //z2// = sqrt{2}(e^(//i//11π/12), and //z3// = sqrt{2}(e^(//i//19π/12)