Convert to Polar and Rectangular Form:

Question

1.) Convert   cos 3pi/4 + isin 3pi/4    into rectangular form. 

 

 

2.) Given:

 

z₁ = 2(cos pi/3 + isin pi/3) and z₂ = -5-5√3 I

 

answer the following questions:

 

a) Express z₂ in polar form

 

b) Find z₁z₂ in polar form

 

c) Find z₁/z₂ in polar form

 

Answer 1

For the first one, rectangular form just means we want the answer to look like a + bi, so all we have to do is evaluate the cosine and the sine separately. The angle 3π/4 sits in the second quadrant, where cosine comes out negative and sine comes out positive, and its reference angle is π/4. So cos 3π/4 is -√2/2 and sin 3π/4 is √2/2, which gives us:

cos 3π/4 + isin 3π/4 = -√2/2 + (√2/2)i

For part a, we are going the other direction now, from rectangular into polar, so we need two things, the modulus r and the angle θ. We get r by using the Pythagorean theorem on the real and imaginary parts, so for z₂ = -5 - 5√3i we have:

r = √((-5)² + (-5√3)²) = √(25 + 75) = √100 = 10

Now for the angle, if you notice both the real part and the imaginary part are negative, which puts this point in the third quadrant, and that is the part students usually miss. The reference angle comes from tan θ = 5√3/5 = √3, so the reference angle is π/3, and since we are in the third quadrant we add that onto π to get θ = π + π/3 = 4π/3. So:

z₂ = 10(cos 4π/3 + isin 4π/3)

For parts b and c, the nice thing about polar form is that multiplying and dividing becomes very easy. When you multiply two complex numbers in polar form, you multiply the moduli and add the angles, and when you divide, you divide the moduli and subtract the angles. That is the whole reason your book has you convert to polar form in the first place, because doing this by FOIL in rectangular form takes much longer. Since z₁ = 2(cos π/3 + isin π/3), for the product we get 2 times 10 = 20 for the modulus and π/3 + 4π/3 = 5π/3 for the angle:

z₁z₂ = 20(cos 5π/3 + isin 5π/3)

And for the quotient we divide 2 by 10 to get 1/5, and subtract the angles to get π/3 - 4π/3 = -π, which we can write as π since those land in the same place:

z₁/z₂ = (1/5)(cos π + isin π)

Remember, if you ever want to check your work on one of these, you can convert your answer back to rectangular form and multiply the original numbers out the long way to see if they match. I did that here, and z₁z₂ comes out to 10 - 10√3i both ways, and z₁/z₂ comes out to -1/5 both ways, which also tells you something interesting, that the quotient landed exactly on the negative real axis.

So, our final answers are -√2/2 + (√2/2)i for the first one, z₂ = 10(cos 4π/3 + isin 4π/3), z₁z₂ = 20(cos 5π/3 + isin 5π/3), and z₁/z₂ = (1/5)(cos π + isin π).