103 Trigonometry Problems

1. Trigonometric Fundamentals 59
or the vector [a,b] that points from the origin to (a, b). Under this convention, the
coordinate plane is called the complex plane. Points (0,b)on the y axis are thereby
matched with pure imaginary numbers bi,sothey axis is called the imaginary axis
in the complex plane. Similarly, the x axis is called the real axis. Let O denote the
origin, and let each lowercase letter denote the complex number assigned to the point
labeled with the corresponding uppercase letter. (For example, if z = 3 + 4i, then
Z = (3, 4).) See Figure 1.58.
The definition of the complex plane allows us to talk about operations on complex
numbers. The sum of the complex numbers z 1 = a 1 + b 1 i and z 2 = a 2 + b 2 i is
z = z 1 + z 2 = (a 1 + a 2 ) + (b 1 + b 2 )i, and their difference is z ′ = z 1 − z 2 =
(a 1 − a 2 ) + (b 1 − b 2 )i. Because of the vector interpretation of complex numbers, it
is not difficult to see that OZ 1 ZZ 2 forms a parallelogram, and z and z ′ are matched
with diagonal vectors −→ OZ and Z −−→
2 Z 1 .
Z = (−2, 6)
z =−2 + 6i
❛
✓❇❇
❛❛❛❛❛❛❛ y ✻
✓ ❇
✓ ❇
❛
✓ ❇
Z 1 = (3, 4)
✓ ❇
z 1 = 3 + 4i
❛❛ ✓ ✓ ✓ ✓
✓
❇
✘✘ ✓
✘✘ ✘✘ ✘ ✘✘✘ ✘ ✘ ✘ ❇
❛❛❛❛❛❛❛✓
Z 2 = (−5, 2) ❇ i
Z ′ = (8, 2)
z 2 =−5 + 2i ❇
z ′ = 8 + 2i
✓✓ ❇✘ ✘ ✘ ✘✘✘ ✘ ✘ ✘ ✘ ✘ ✘✘ ✘
✲
Q = (−5.5, 0) O
x
q =−5.5 + 0i
P = (−2, −3)
p =−2 − 3i
S = (2, −3.5)
s = 2 − 3.5i
Figure 1.58.
We can talk about the magnitude or length of a complex number z, denoted
by |z|, by considering the magnitude of the vector it corresponds to. For example,
|3−4i| =5, and in general, |a +bi| = √ a 2 + b 2 . Hence, all the complex numbers z
with |z| =5 form a circle centered at the origin with radius 5. We can also talk about
the polar form of complex numbers. If Z = (a, b) has polar coordinates Z = (r, θ),