Math 5, Trig

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Mattyj1388
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117791
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Math 5, Trig
Updated:
2011-11-19 13:08:07
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Chapter8 Math 005 COD Trig polar coordinates
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  1. 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/3-2pi) = (2,-5pi/3)
    • 3. (-2, pi/3+pi) = (-2, 4pi/3)
    • 4. (-2, pi/3- pi) = (-2, -2pi/3)
  2. What is the polar equation for x?
    x = r cos Theta
  3. What is the polar equation for y?
    y = r sin Theta
  4. What is the polar equation for y / x?
    y/x = tan Theta (where x cannot = 0)
  5. What is the rectangular equation for r2?
    r2 = x2 + y2
  6. 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














































































  7. 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
  8. 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
  9. What does symmetry about the y axis look like?
    ie: (pi/2)
  10. 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.
  11. What does Lemniscates look like?
    • r2 = a2 cos (2 Theta)
    • r2 = a2 sin(2 Theta)
  12. 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.
  13. What is the polar form of a + bi?
    z = r ( cos theta + i sin theta)
  14. What is DeMoivre's theorem for :
    z = r ( cos theta + i sin theta)?
    zn = r n( cos theta + i sin theta)
  15. What is the nth roots of complex numbers for:
    z = r (cos theta + i sin theta)?
    • If n is a positive intiger, than:
    • wk = r1/n [cos (theta + 2kpi)/n + i sin (theta +2kpi)/n]
    • for k = 0, 1, 2, 3. . . . . n-1.
  16. What are the general formulas for adding, subtracting and multiplying by a constant to vectors? given:
    u = <a1,b1> and v = <a2,b2>
    • u + v = <a1+a2,b1+b2>
    • u - v = <a1-a2,b1-b2>
    • cu = <ca1,cb1>, c = real number
  17. Addition Properties of Vectors.
    • Addition:
    • u + v = v + u
    • u + (v + w ) = ( u + v ) + w
    • u + 0 = u
    • u + (-u) = 0
  18. Length Propertie for Vectors
    | cu | = | c| |u |
  19. 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
  20. How is x represented in vector form?
    x = i = < 1 , 0 >
  21. How is y represented in vector form?
    y = j = < 0, 1 >
  22. 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
  23. Dot product:
    Given u = < a1 , b1 >
    and
    v = < a2 , b2 >
    u * v = a1a2 + b1b2
  24. 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
  25. The dot product theorem
    u * v = | u | |v | cos theta
  26. Angle between two vectors
    cos theta = ( u * v) / [ | u| | v | ]
  27. Orthoganol Vectors
    • two nonzero vecters are perpendicular when their dot product is 0:
    • u * v = 0
  28. The component u along v is:
    ( u * v ) / | v |
  29. The projection of u onto v is the vector projv u given by
    • projv u = [ ( u *v ) / | v |2 ] v
    • If the vector u is resolved into u1 and u2, where u1 is parallel to v and u2 is orthagonal to v, then
    • u1 = projvu and u2 = u - projvu
  30. Work
    • W = F * D
    • W= work
    • F= force
    • D= vector

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