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List 4 other polar coordinates for P(r,Theta) where
r = 2; Theta = pi / 3 or (2, pi/3).
 1. (2, pi/3 +2pi) = (2, 7pi/3)
 2. (2,pi/32pi) = (2,5pi/3)
 3. (2, pi/3+pi) = (2, 4pi/3)
 4. (2, pi/3 pi) = (2, 2pi/3)

What is the polar equation for x?
x = r cos Theta

What is the polar equation for y?
y = r sin Theta

What is the polar equation for y / x?
y/x = tan Theta (where x cannot = 0)

What is the rectangular equation for r^{2}?
r^{2} = x^{2 }+ y^{2}

What are all the values given r = 2 sin Theta for the values of Theta being:
0, pi/6, pi/4, pi/3, pi/2, 2pi/3, 3pi/4, 5pi/6, and pi






Θ 
0 
π/6 
π/4 
π/3 
π/2 
2π/3 
3π/4 
5π/6 
π 
2 sinΘ 
0 
1 
√(2) 
√(3) 
2 
√(3) 
√(2) 
1 
0 

What does a cardioid look like? What is the equation for a cardioid?
 r = a +or  b cos theta
 or
 r = a +or b sin theta

What does symmetry about the y axis look like?
ie: (pi/2)

What does a Limacons look like?
 a > b ; a = b a < b
 The bigger the constant in front of sin gets, the bigger the loop gets inside the circle.

What does Lemniscates look like?
 r^{2} = a^{2} cos (2 Theta)
 r^{2} = a^{2} sin(2 Theta)

What is the equation for a rose?
 r = a cos n Theta
 n = even #, then there will be 2n number of leaves.
 n = odd #, then there will be n number of leaves.

What is the polar form of a + bi?
z = r ( cos theta + i sin theta)

What is DeMoivre's theorem for :
z = r ( cos theta + i sin theta)?
z^{n} = r ^{n}( cos theta + i sin theta)

What is the nth roots of complex numbers for:
z = r (cos theta + i sin theta)?
 If n is a positive intiger, than:
 w_{k} = r^{1/n} [cos (theta + 2kpi)/n + i sin (theta +2kpi)/n]
 for k = 0, 1, 2, 3. . . . . n1.

What are the general formulas for adding, subtracting and multiplying by a constant to vectors? given:
u = <a_{1},b_{1}> and v = <a_{2},b_{2}>
 u + v = <a_{1}+a_{2},b_{1}+b_{2}>
 u  v = <a_{1}a_{2},b_{1}b_{2}>
 cu = <ca_{1},cb_{1}>, c = real number

Addition Properties of Vectors.
 Addition:
 u + v = v + u
 u + (v + w ) = ( u + v ) + w
 u + 0 = u
 u + (u) = 0

Length Propertie for Vectors
 cu  =  c u 

Multiplication Properties of Vectors.
 Multiplication:
 c (u + v ) = cu + cv
 ( c + d ) u = cu + du
 (cd) u = c (du) = d (cu)
 1u = u
 0u = 0
 c 0 = 0

How is x represented in vector form?
x = i = < 1 , 0 >

How is y represented in vector form?
y = j = < 0, 1 >

Horizontal and vertical components of a vector.
 magnitude =  v  in the direction of theta.
 Then v = < a, b > = ai + bj, where
 a =  v  cos theta i and b =  v  sin theta j
 thus, we can express v as:
 v =  v  cos theta i +  v  sin theta j

Dot product:
Given u = < a_{1} , b_{1 }>
and
v = < a_{2} , b_{2} >
u * v = a_{1}a_{2} + b_{1}b_{2}

Properties of dot product.
 u * v = v * u
 (au) * v = a (u*v) = u * (av)
 ( u + v) * w = u * w + v * w
  u ^{2} = u * u

The dot product theorem
u * v =  u  v  cos theta

Angle between two vectors
cos theta = ( u * v) / [  u  v  ]

Orthoganol Vectors
 two nonzero vecters are perpendicular when their dot product is 0:
 u * v = 0

The component u along v is:
( u * v ) /  v 

The projection of u onto v is the vector proj_{v }u given by
 proj_{v }u = [ ( u *v ) /  v ^{2} ] v
 If the vector u is resolved into u_{1} and u_{2}, where u_{1} is parallel to v and u_{2} is orthagonal to v, then
 u_{1} = proj_{v}u and u_{2} = u  proj_{v}u

Work
 W = F * D
 W= work
 F= force
 D= vector


