Download 8.4 * Trigonometric Form of Complex Numbers

8.4 – Trigonometric Form of
Complex Numbers
• From a while back, we defined a complex
number as a number that may be written as…
– z = a + bi
– a is the real part
– b is the imaginary part (not bi…that drives me
crazy!)
Graphing Complex Numbers
• Graphing complex numbers is similar to
graphing numbers in the Cartesian plane
• Horizontal = real
• Vertical = imaginary
• Example. Graph the imaginary number z = 4 –
3i
• Example. Graph the imaginary number z = 2 +
2i
Magnitude
• A complex number is similar to a vector, in
that we may find a magnitude or modulus of
a complex number
• For the complex number, z = a + bi
• |z| =
a b
2
2
• Example. Determine the magnitude of the
complex numbers:
• A) -2 + 5i
• B) 3 + 4i
• C) -9i
Trig Form of Complex Numbers
• In the case of complex numbers, a lot of times
their actual form as a complex number may
not be useful
• Luckily, we have a way to convert a complex
number back to a real number
• Extremely useful in helping to use complex
numbers back in terms of parabolas, trig
equations, etc.
Conversion
• If z = a + bi, then the imaginary number z may
be rewritten as…
• z = |z| (cosϴ + isinϴ)
• ϴ is such that tanϴ = b/a
– The angle is known as the “argument”
• Example. Write the complex number z = 3 + i
in trigonometric form.
• Example. Write the complex number 5 - 2i in
trigonometric form.
• Assignment
• Pg. 654
• 1-6, 17-28