Some Solutions for 6.5 Trigonometric Form of a Complex Number
1. Find the 8 roots of unity, that is, solve x8=1
roots are: 1,
√2
2
+
√2
𝑖
2
, 𝑖,
−√2
2
+
√2
𝑖,
2
-1,
−√2
√2
− 2 𝑖,
2
−𝑖,
√2
2
−
√2
𝑖
2
2. Prove |z1z2| = |z1|*|z2|
|(𝑎 + 𝑏𝑖)(𝑐 + 𝑑𝑖)| = |(𝑎𝑐 − 𝑏𝑑) + (𝑎𝑑 + 𝑏𝑐)𝑖| =
√(𝑎𝑐 − 𝑏𝑐)2 + (𝑎𝑑 + 𝑏𝑐)2 = √𝑎2 𝑐 2 + 𝑏 2 𝑑2 + 𝑎2 𝑑2 + 𝑏 2 𝑐 2 = √𝑎2 (𝑐 2 + 𝑑2 ) + 𝑏 2 (𝑐 2 + 𝑑2 ) =
√(𝑎2 + 𝑏 2 )(𝑐 2 + 𝑑2 ) = √(𝑎2 + 𝑏 2 ) · √(𝑐 2 + 𝑑2 ) = |z1|*|z2|
3. One can use the quadratic formula
z =( -b + (b2-4ac)1/2)/2a
where a, b and c are the complex numbers. Both square roots are to be considered when discriminant is
non-zero.
-
find the roots of the equation z2 + 2z + (1 – i) = 0
-
Solutions: −1 +
-
find the roots of the equation z2 – 4z + (4 + 2i) = 0
Solutions: 3 − 𝑖 and 1 + 𝑖
√2
2
+
√2
𝑖
2
and −1 −
√2
2
−
√2
𝑖
2
4. Use properties of conjugates and moduli to show that
a) z  3i  z  3i
b) iz  iz
c)
(2  i ) 2  3  4i
d)
(2 z  5)( 2  i )  3 2 z  5 hint: use property from #2 on this w.s.
5. Prove that
a) Z is real if and only if z  z
b) Z is either real or pure imaginary if and only if ( z ) 2  z 2