Chapter 9 - Polar Coordinates

Expand (1 + i)^12.

DeMoivres Method [√2 (cos π/4 + i sin π/4)]^12 = (√2)^12(cos π/4 + i sin π/4)^12 = 2^6 (cos 3π + i sin 3π) = 64(cos π + i sin π) = 64(-1) = ANSWER: -64.

i pattern

Every 4: i, -1, -i, 1 i^16= 1

Polar coordinates to Rectangular coordinates

Example: A point has polar coordinates: (5, 30º). Convert to rectangular coordinates. Solution: (x,y) = (5cos30º, 5sin30º) = (4.3301, 2.5)

Rectangular coordinates to Polar coordinates

So the rectangular point: (x,y) can be converted to polar coordinates like this: (√x^2+y^2), tan-1( y/x ) ) ð ( r , q ) Example: A point has rectangular coordinates: (3, 4). Convert to polar coordinates. Solution: r = square root of(3² + 4²) = 5, q = tan-1(4/3) = 53.13º so (r,q) = (5, 53.13º)

Put 2x+y-5=0 into normal form

[(2√5)/5]x+(√5/5)y-√5=0

Solve z^5+ 32 = 0

= 32^1/5[cos(π/5+2πk/5)+isin(π/5+2πk/5)] 5 answers #1: = 2[cos(π/5)+isin(π/5)] ANS: 1.618+.588i #2: = 2[cos(3π/5)+isin(3π/5)] #3 = -2 #4 = 2[cos(7π/5)+isin(7π/5)] #5 = 2[cos(9π/5)+isin(9π/5)]

Expand (√3 + i)^5

= [2(cos π/6 + i sin π/6)]^5 = 2^5(cos π/6 + i sin π/6)^5 = 32(cos 5π/6 + i sin 5π/6) = 32(-√3/2 + 1/2 i) = -16√3 + 16 i.

Polar form to Rectangular form

Change r = -3 cos (θ) to rectangular form Solution: Use: r^2 = x^2 + y^2 and x = r cos (θ) r = - 3 cos (θ) [Multiply by r to get r^2] r^2 = -3r cos (θ) [Use r^2= x^2+ y^2] x^2+ y^2 = - 3r cos (θ) [ Use x = r cos (θ)]

Rectangular form to polar form

Change x^2 + y^2 - 2y = 0 to polar form Solution : Use: r^2 = x^2 + y^2 and y = r sin(θ) x^2 + y^2 - 2y = 0 [Replace (x2 + y2 ) with r2 ] r^2 - 2y = 0 r^2 - 2( r sin(θ) ) [replace y with r sin(θ)] r (r - 2sin(θ)) = 0 [factor out r ] r - 2sin (θ) = 0 ANSWER: r = 2sin (θ)

Write the polar coordinates for (-3,4)

r^2=4^2+3^2...r=√25)........... tanØ=-4/3 (5,126.9)

x^2+y^2=3x

r^2cos^20+r^2sin^20=3rcos0 r^2=3rcos0 r=3cos0 0=feta

y=-2

rsin0=-2 (Divide each side by sin0) r=-2csc0 0=feta

Conversion: Rectangular to Polar/ Polar to Rectangular

x2+ y2 + 3x = 0 [Rectangular form ] (x2+ 3x) + y2= 0 reorganize in x2 + y2 = r2 ( x2+ 3x + 9/4)+ y2= 0 +9/4 [complete the square] (x + 3/2)^2+ y^2= 9/4 [rectangular form] Ex: given point = 4 at 30degree. Convert to rectangular: y = r sin(θ), x = r cos(θ) so [x, y] = [4cos(30), 4sin(30)] = ANSWER: [2√3, 2]