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| =

Example. Determine the magnitude of the complex numbers: A) i 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 , 17-28