Section 10.3 The Complex Plane; De Moivre's Theorem. abi

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Ernest Carroll
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1 Sectio 03 The Complex Plae; De Moivre's Theorem REVIEW OF COMPLEX NUMBERS FROM COLLEGE ALGEBRA You leared about complex umbers of the form a + bi i your college algebra class You should remember that "i" is the imagiary uit ad it is defied as i By squarig both sides, e see that i I a complex umber a + bi, a is called the part ad b is called the part You should also recall that a bi is called the of a + bi If e multiply a complex umber by its cojugate, e get the folloig result: a bi a bi a abi abi 5i 5i b i a b a b For example, I your college algebra class, you ever leared ho to graph complex umbers We ill do that o THE COMPLEX PLANE We ill o defie a complex umber as "", here x yi replaces our typical a + bi otatio Therefore, x becomes the real part, hich e ill plot o the real axis, ad y is the imagiary part, hich e ill plot o the imagiary axis The distace from the origi to the poit is called the magitude of Usig the distace formula d x x y y e fid that the magitude 0 0 x y x y Sice x yi, e ill deote its cojugate as (called "-bar") The product of ad is: x yi x yi x y x y Example : If 3 i, Graph ad fid ad CONVERTING A COMPLEX NUMBER FROM RECTANGULAR FORM TO POLAR FORM Recall that x r cos ad y r si Therefore, e ca rite a complex umber as ( r cos ) ( r si ) i, ad if e factor out the r, e ca also rite it as r cos isi Recall also that r x y, so r x y No, sice the magitude of is x y, that meas that r Lastly, recall that ta y x, so y ta x 03 Page

2 Example : Plot each complex umber i the complex plae ad rite it i polar form Express the argumet i degrees a) i Start by plottig the poit i the complex plae Sice x = ad y =, this poit is i Quadrat b) 3i No e fid r ad θ: r r i Q ta ta 35 Thus, the complex umber i is equivalet to cos35 isi35 i polar form Start by plottig the poit i the complex plae Sice x = ad y =, this poit is i Quadrat No e fid r ad θ: r 3 ta Thus, the complex umber 3i is equivalet to = i polar form Example 3: Plot the poit i the complex plae ad rite each complex umber i rectagular form a) cos0 isi0 Recall that a complex poit i polar coordiates has the form r cos isi, so for this problem, r = ad θ = 0 This poit is i Quadrat Note that cosie ad sie are both i this quadrat ad 0 has a referece agle of Thus, cos0 cos30 ad si0 si30 Substitutig these values ito the origial poit gives: Distribute the cos0 isi0 i 03 Page

Example 3 (cotiued): Plot the poit i the complex plae ad rite each complex umber i rectagular form b) 4cos i si For this problem, r = ad θ = This is a agle so e eed to use the to fid the values of

3 Example 3 (cotiued): Plot the poit i the complex plae ad rite each complex umber i rectagular form b) 4cos i si For this problem, r = ad θ = This is a agle so e eed to use the to fid the values of cos = ad si = So 4 i FINDING PRODUCTS AND QUOTIENTS OF COMPLEX NUMBERS IN POLAR FORM We ill use these formulas to fid a product or quotiet of complex umbers ritte i polar form These formulas are derived by multiplyig by ad the applyig the sum/differece formulas e leared back i sectio 84 Example 4: Fid ad Leave your asers i polar form a) cos0 isi0, cos00 isi00 For both ad, r = 0 ad 00 cos si cos0 00 isi0 00 cos0 isi0 r r i r cos r isi cos 0 00 i si 0 00 cos0 isi0 b) cos i si, cos i si 8 8 r,, r, 03 Page 3

USING DE MOIVRE'S THEOREM De Moivre's Theorem allos us to fid poers of complex umbers usig the formula o the right It is a very simple formula to use Let's look at to examples Example 5: Write each

4 USING DE MOIVRE'S THEOREM De Moivre's Theorem allos us to fid poers of complex umbers usig the formula o the right It is a very simple formula to use Let's look at to examples Example 5: Write each expressio i the stadard form a + bi a) cos isi I this case, 5 4, r, ad cos 4 i si 4 4 cos isi 4 i i So 4 b) cos isi =, r =, ad θ = So 6 c) 3 i 6 We first have to rite this complex umber i polar form Sice x 3 ad y, this poit is i Q Recall that r x y So y ta ta x 3 r Sice θ must be i Q4, the So 3 i is equivalet to cos i si i polar form cos 6 si 6 cos 6 i si The r i i cos is eve, si is odd i i 64 cos si 64 cos si Page 4

FINDING COMPLEX ROOTS Here e lear ho to fid roots of complex umbers Note that if you are orkig i degrees, the you ould use 360 istead of, so the formula becomes: 0 360 0 360 k k k r cos isi, here I

5 FINDING COMPLEX ROOTS Here e lear ho to fid roots of complex umbers Note that if you are orkig i degrees, the you ould use 360 istead of, so the formula becomes: k k k r cos isi, here I chose to rerite r as r This is ot the clearest formula i the orld, so let's look at a example to see ho it orks Example 6: Fid all the complex cube roots of 8 8i Leave your asers i polar form ith the argumet i degrees First e express the umber i polar form usig degrees With x = 8 ad y = 8, this umber is i Q r 8 I Q3 ta ta 5 8 Because e are fidig "cube" roots, use = 3 i the formula: k 0 360k k r cos isi k k 5 360k cos i si k cos 75 0k isi 75 0k Multiply the expoets together No e evaluate for k = 0,, ad k 0 : 0 cos i si cos75 i si75 k : cos 75 0 isi 75 0 cos95 isi95 k : cos 75 0 i si 75 0 cos35 i si35 03 Page 5

APPENDIX F Complex Numbers

APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios