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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