Download SECTION 7-6 Complex Numbers in Rectangular and Polar Forms

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7 Additional Topics in Trigonometry
EXAMPLE 2
From Rectangular to Polar Form
Write parts A–C in polar form, in radians, . Compute the modulus
and arguments for parts A and B exactly; compute the modulus and argument for part
C to two decimal places.
(A) z1 1 i
Solution
(B) z2 3 i
(C) z3 5 2i
Locate in a complex plane first; then if x and y are associated with special angles, r
and can often be determined by inspection.
(A) A sketch shows that z1 is associated with a special 45° triangle (Fig. 4). Thus,
by inspection, r 2, /4 (not 7/4), and
4 i sin 4 z1 2 cos 2e(/4)i
FIGURE 4
y
1
x
1
r
1i
(B) A sketch shows that z2 is associated with a special 30°–60° triangle (Fig. 5).
Thus, by inspection, r 2, 5/6, and
z2 2 cos
5
5
i sin
6
6
2e(5/6)i
y
FIGURE 5
3 i
1
r
3
x
(C) A sketch shows that z3 is not associated with a special triangle (Fig. 6). So, we
proceed as follows:
r (5)2 (2)2 5.39
To two decimal places
tan
To two decimal places
1
2
5
2.76