# Math 5, Trig

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 Author: Mattyj1388 ID: 117791 Filename: Math 5, Trig Updated: 2011-11-19 13:08:07 Tags: Chapter8 Math 005 COD Trig polar coordinates Folders: Description: General Show Answers:

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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
• 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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