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Pre Calc 6.5 Day 1 Trigonometric Form of a Complex Number NOTES COMPLETED 6.5 Trigonometric Form of a Complex Number A complex number looks like a+ bi a.k.a. Plot (2,3) in the coordinate plane Plot (1,2) 3 2 1 2 11 2 1 2 3 Absolute Value Real Number 1. 5 = 2. 5 = 3. Find 2 + 5i = Plot 2+ 3i in the coordinate plane Plot 12i Imaginary numbers always appear in pairs a ± bi Find the conjugates: Horizontal axis REAL axis 1. 2+ 3i _______ Vertical 2. 5 6i _______ axis IMAGINARY axis 3. 2 + 3i _______ 4. 7 _______ 5. 3i ______ Complex Number: Distance between (0,0) and point (a,b). The absolute value of the complex number a + bi is Trig Form of a Complex Number r θ y x r = x 2 + y2 y tan θ = ___ x cos θ = adj sin θ = opp hyp hyp (a,b) cos θ = sin θ = x = y = (x,y) . Radius of Circle Trig Form of a Complex Number z = a + bi or z = x + yi is z = a + bi z = x + y i z = r cosθ + r sinθ i z = r(cos θ + i sin θ) You do not r is called the ________ of z. Modulus have to is called an _________ of z. Argument know for θ test Write the complex number in TRIG FORM: 1. z = 2 2 √3 i or Answers written 3 ways using radians using degrees shortcut way using radians 2) 6 6i 4 3. z = 3√ 3 +3i 4. 7+ 0i x= y= r= cos = sin = x= y= r= cos = sin = 0 0 5. z = 4 (cos120 + i sin120 ) 1 6. z = (cos35 + i sin35 ) 0 2 0 Note: if the degree/radian given is off the Trig Circle, you can use decimals!! woohoo!! Type the entire problem into your calculator and don't forget to round to 2 places. Homework 6.5 pg 479: 1123 odd [ 3143 odd[ WPF/skip all graphs/skip #19 write all 3 formats of answer WPF/skip all graphs find answers like pg 3 notes Pre Calc 6.5 Day 2 Trigonometric Form of a Complex Number NOTES COMPLETED 6.5 Multiplication and Division of Complex Numbers [ x and + and z1z2= your work should keep 0 within the interval [0,2 ) ll z1 z 2 = 4 Remember: numbers need to be in the interval [0,2 ) ll = 1 distance from zero Use DeMoivre's Theorem to simplify [We are not going to use DeMoivre's Thm. Just use your grapher] 8i (1 + i)6 = ____ 2035 828 32 3 16 4096 5) (1 + 3 i)12 = _______ 4 6) (1 + i)4 = ____ Note: 4 8E13 = 4 0 = 4 Today's Homework 6.5 pg 479: 4755 odd (AO). Keep original units. Don't solve down to standard form. Stop at the line before you get rid of the cos and sin. Example of how to do problems 4755 odd pg 479: 5966 all (part c only) Simplify. Show all work and write answers in standard form