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Trigonometric Form of Complex Numbers Lesson 5.2 Graphical Representation of a Complex Number Graph in coordinate plane Called the complex plane Horizontal axis is the real axis Vertical axis is the imaginary axis 3 + 4i • -2 + 3i • • -5i 2 Absolute Value of a Complex Number Defined as the length of the line segment From the origin To the point Calculated by using Pythagorean Theorem 3 + 4i • 3 4i 32 42 25 5 3 Find That Value, Absolutely Try these Graph the complex number Find the absolute value z 4 4i z 5 z 5 6i 4 Trig Form of Complex Number Consider the graphical representation We note that a right triangle is formed a + bi • r b θ a a cos r a r cos b sin r b r sin where r z a 2 b 2 How do we determine θ? b tan a 1 5 Trig Form of Complex Number Now we use a r cos b r sin and substitute into z = a + bi z r cos i r sin Result is Abbreviation is often z r cis 6 Try It Out Given the complex number -5 + 6i Write in trigonometric form r=? θ=? Given z = 3 cis 315° Write in standanrd form r=? a=? b=? 7 Product of Complex Numbers in Trig Form Given z1 r1 cos1 i sin 1 z2 r2 cos2 i sin 2 It can be shown that the product is z1 z2 r1 r2 cis 1 2 Multiply the absolute values Add the θ's 8 Quotient of Complex Numbers in Trig Form Given z1 r1 cos1 i sin 1 z2 r2 cos2 i sin 2 It can be shown that the quotient is z1 r1 cis 1 2 z2 r2 9 Try It Out Try the following operations using trig form 4 cis120 6 cis315 15cis240 3cis135 Convert answers to standard form 10 Assignment Lesson 5.2 Page 294 Exercises 1 – 57 odd 11