Amy's Reflection #31

here's some stuff from chapter 11..

Imaginary Numbers are no longer "imaginary"

Rectangular form: a + bi

Polar form: z = r cos theta + r sin theta i (abbreviated z = r cis theta)

Examples:

1. Express 2 cis 50degrees in rectangular form

2 cos 50 + 2 sin 50 i

2. Express -1-2i in polar form

radius = +- sqrt of ((-1)^2 + (-2)^2)) = +- sqrt of (5)

theta = tan^-1(-2/-1)

theta = tan^-1(1)

*tangent is positive in the first and third quadrants, 63.435 and 243.435
*63 is positive for cosine so it goes with the positive sqrt of 5
*243 is negative for cosine so it goes with the negative sqrt of 5

z= sqrt of 5 cis 63.435

z= sqrt of 5 cos 63.435 + sqrt of 5 sin 63.435 i

z= negative sqrt of 5 cis 243.435

z= negative sqrt of 5 cos 243.435 + negative sqrt of 5 sin 243.435 i

De Moivre's Theorem: z^n = r^n cis(n)(theta)

Examples:

1. z=2cis20degrees Find z^2

z^2=2^2cis2(20degrees)

z^2=4cis40degrees

2. 4cis15degrees Find z^4

z^4=4^4cis4(15degrees)

z^4=256cis60degrees

Limacon
r=a+b sin(theta)
r=a+b cos(theta)

Cardioid
a-b sin(theta)
r=a-b cos(theta)

Rose
r=a sin(n theta)
r=a cos (n theta)

*n=how many petals

Archimedes Spiral
r=a theta+b

Logarithmic Spiral
r=a b^theta

Examples:
1. r=theta+2
2. r=2+3cos(theta)
3. r=5
4. r=3sin(4 theta)
5. r=1/2(3^theta)
6. r=2sin(theta)

1. archimedes spiral
2. limacon
3. circle with its center at the pole
4. rose with 4 petals
5. logarithmic spiral
6. circle that intersects with the pole

ok what i really dont understand is the first two sections..if someone could explain them to me that would be awesome..thanks..