we this week was all about chapter 11 so here are some of the stuff we went over & examples..
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..
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Chapter 11 sections 1 and 2 were pretty much an introduction to polar. They basicly gave you all of the equations you will need for polar. They tell you how to convert from rectangular to polar and vice versa. When they tell you to plot, or just give you a point in polar(with an x and an angle), you then draw your x-axis with only positive numbers and plot them positively, then you trace the distance to the opposite side of the axis, only if negative though.
ReplyDeletethe first two sections are far easier than the following two sections
ReplyDeletethe first section is just a matter of drawing an arrow and picturing a protractor where the given degree would be and draw a dot at that point then just write the degree next to it
for negatives do the same but put a second pt diagonally from the first on the opposite side of the arrow and circle it
besure to write the degree next to the circled pt.
the second section can be simplified into two methods
pol to rec is plugging into
X= rcos(degree)
Y= rsin(degree)
after plugging in and solving completely place into (X,Y)
rec to pol is plugging into
r= +/- squareroot X^2 + Y^2
(theta)= tan inverst (Y/X)
then plug answers into
(+r, theta)
(-r,theta)