Lesson 7 Polar Form of Complex Numbers HL Math - Santowski Relationships Among x, y, r, and x r cos y r sin r x y tan y x, if x 0 Polar Form of a Complex Number The expression r(cos isin ) is called the polar form (or trigonometric form) of the complex number x + yi. The expression cos + i sin is sometimes abbreviated cis. Using this notation r(cos isin ) is written r cis. 1
Example Express (cos 10 + i sin 10) in rectangular form. Example Express (cos 10 + i sin 10) in rectangular form. 1 1 cos10 (cos10 isin10 ), i 1 i sin10 Notice that the real part is negative and the imaginary part is positive, this is consistent with 10 degrees being a quadrant II angle. Converting from Rectangular to Polar Form Step 1 Sketch a graph of the number x + yi in the complex plane. Step Find r by using the equation r x y. Step y Find by using the equation tan, x 0 x choosing the quadrant indicated in Step 1.
Example Example: Find trigonometric notation for 1 i. Example Example: Find trigonometric notation for 1 i. First, find r. r a b r ( 1) ( 1) r 1 1 sin cos 5 4 5 5 5 Thus, 1 i cos isin or cis 4 4 4 Product of Complex Numbers Find the product of 4(cos50 isin50 ) and (cos10 isin10 ).
Product Theorem If r1 cos1 isin 1 and r cos isin, are any two complex numbers, then cos sin cos sin cos isin. r i r i 1 1 1 rr 1 1 1 In compact form, this is written r r rr cis cis cis. 1 1 1 1 Example: Product Find the product of 4(cos50 isin50 ) and (cos10 isin10 ). Example: Product Find the product of 4(cos50 isin50 ) and (cos10 isin10 ). 4(cos50 isin50 ) (cos10 isin10 ) 4 cos(50 10 ) isin(50 10 ) 8(cos60 isin60 ) 8 1 i 4 4i 4
Quotient of Complex Numbers Find the quotient. 16(cos70 isin70 ) and 4(cos40 isin 40 ) Quotient Theorem If r cos isin and r cos isin 1 1 1 are any two complex numbers, where r cos isin, r 0, then r1 cos1 isin1 r1 cos1 isin 1. r cos isin r In compact form, this is written r1 cis 1 r1 cis1 r cis r Example: Quotient Find the quotient. 16(cos70 isin70 ) and 4(cos40 isin 40 ) 5
Example: Quotient Find the quotient. 16(cos70 isin70 ) and 4(cos40 isin 40 ) 16(cos70 isin70 ) 4(cos40 isin 40 ) =16 4 cos(70 40 ) isin(70 40 ) 4cos0 isin0 4 1 i i cos 40 i sin 40 and w 6cos10 i sin10 If z 4, find : (a) zw (b) z w If z 4cos 40 o isin40 o and w 6cos10 o isin10 o, find: (a) zw (b) z w zw 4 cos 40 i sin 40 6 cos10 i sin 10 4 6cos40 10 i sin40 10 multiply the moduli 4 0.9969.55 8.1i 4 cos160 i sin 160 add the arguments (the i sine term will have same argument) 0.40i If you want the answer in rectangular coordinates simply compute the trig functions and multiply the 4 through. 6
z w 4 cos 40 o isin40 o 6 cos10 o isin10 o 4 6 cos 40o 10 o isin 40 o 10 o divide the moduli cos 80o isin 80 o In rectangular coordinates: cos 80oisin 80 o subtract the arguments In polar form we want an angle between 0 and 60 so add 60 to the -80 0.176 0.9848i 0.1 0. 66i z1 divide z Example Where z1 i 6cos45 isin45 z i 4cos0 isin0 Rectangular form Trig form z1 divide z Example Where z1 i 6cos45 isin45 Rectangular form i i i i i i 6 6 6 i 6 6i 6 i 1 4i 6 6 6 i 6 6i 6 1 4 6 6 6 6 6 6 i 16 16 z i 4cos0 isin0 Trig form 6cos45 isin45 4cos0 isin0 6 cos 45 0 i sin 45 0 4 cos15 i sin15 6 6 6 6 6 6 6 6 6 r 1 16 16 tan 16 15 6 6 6 16 7 1 7 16 7 1 7 r 16 56 576 9 r 56 4 7
DeMoivre s Theorem Taking Complex Numbers to Higher Powers z r cos i sin cos sin cos sin z?? r i r i r cos icos sin i sin r cos r z r sin isin cos cos isin cos i sin cos sin cos sin cos sin z r i z r i r i r cos icos sin i sin r r cos sin isin cos cos isin z r cos i sin Nice 8
What about z? cos isin, cos sin z z r cos i sin r cos i sin z r z r i z z r i i r r r cos cos sin sin cos sin isin cos cos isin cos isin cos sin cos sin r i i r cos cos i cos sin i sin cos i sin sin r i n isin cos i s cos sin sin cos sin cos cos sin sin cos si r cos cos sin sin sin co r cos i r i z r cos isin Hooray!! 9
We saw z r cos isin and z r cos i Similarly. 4 4 z r cos4 isin 4 5 5 z r i cos5 sin 5 sin n n z r cosn isin n We saw z r cos isin and z r cos i Similarly. 4 4 z r cos4 isin 4 5 5 z r i sin cos5 sin 5 n n z r cosn isin n Hooray DeMoivre and his incredible theorem Powers of Complex Numbers This is horrible in rectangular form. n a bi a bia bia bi... a bi The best way to expand one of these is using Pascal s triangle and binomial expansion. You d need to use an i-chart to simplify. It s much nicer in trig form. You just raise the r to the power and multiply theta by the exponent. z r cos i sin n n z r cosn i sinn Example z 5cos0 isin0 z 5 cos0 isin0 z 15cos60 isin60 10
De Moivre s Theorem If r cos isin 1 1 1 is a complex number, and if n is any real number, then n n r cos 1 isin 1 r cos n isin n. In compact form, this is written n n r r n cis cis. Example: Find (1 i) 5 and express the result in rectangular form. Example: Find (1 i) 5 and express the result in rectangular form. First, find trigonometric notation for 1 i 1 i cos 5 isin 5 Theorem 5 1 i cos5 isin 5 5 cos( 5 5 ) isin( 5 5 ) 4 cos115 i sin115 4 i 44i 5 11
Example z 1 i Let. Find z 10 Example Further Examples 1
nth Roots For a positive integer n, the complex number a + bi is an nth root of the complex number x + yi if a bi x yi. n nth Root Theorem If n is any positive integer, r is a positive real number, and is in degrees, then the nonzero complex number r(cos + i sin ) has exactly n distinct nth roots, given by n where n r cos isin or r cis, 60 k 60 k or =, k 0,1,,..., n1. n n n Example: Square Roots i Find the square roots of 1 1
Example: Square Roots Find the square roots of 1 Polar notation: For k = 0, root is For k = 1, root is 1 i i cos60 isin60 1 1 cos60 isin60 cos 60 k 60 isin 60 k 60 cos 0 k 180 isin 0 k 180 cos0 isin0 cos10 isin10 Example: Fourth Root Find all fourth roots of 8 8i. Example: Fourth Root Find all fourth roots of 8 8i. roots in rectangular form. Write the Write in polar form. 8 8i 16 cis 10 Here r = 16 and = 10. The fourth roots of this number have absolute value 4 16. 10 60 k 0 90 k 4 4 14
Example: Fourth Root continued There are four fourth roots, let k = 0, 1, and. k 0 0 90 0 0 k 1 0 90 1 10 k 0 90 10 k 0 90 00 Using these angles, the fourth roots are cis 0, cis 10, cis 10, cis 00 Example: Fourth Root continued Written in rectangular form i 1i i 1 i The graphs of the roots are all on a circle that has center at the origin and radius. Find the complex fifth roots of 15
Find the complex fifth roots of The five complex roots are: for k = 0, 1,,,4. Example Find all the complex fifth roots of 16
0i r a b 1 0 0 tan tan 1 0 0 i sin cos 0 0 5 0 60k 0 60k cos i sin 5 5 5 5 cos 7k i sin 7k k 0 cos 0 i sin0 1 0i k 1 cos 7k i sin 7k i sin cos 7 7. 6 1. 90i k cos 144 i sin144 1. 6 1. 18i k cos 16 i sin16 1. 6 1. 18i k 4 cos 88 i sin88. 6 1. 90i Example Find i You may assume it is the principle root you are seeking unless specifically stated otherwise. First express i as a complex number in standard form. 0 i Then change to polar form r a b r 1 b tan a 1 tan 0 i sin 1cos 90 90 tan 90 0 1 1 17
i sin 1cos 90 90 Since we are looking for the cube root, use DeMoivre s Theorem 1 and raise it to the power 1 1 1 1 cos 90 i sin 90 i sin 1 cos 0 0 1 1 i 1 i Example: Find the 4 th root of i i 1 4 Change to polar form 5 cos 15. 4 i sin 15. 4 Apply DeMoivre s Theorem 1 4 1 1 5 cos 15. 4 i sin 15. 4 4 4 1. cos 8. 4 i sin 8. 4. 96. 76i 18