## anonymous one year ago Transform each polar equation to an equation in rectangular coordinates and identify its shape. a. θ = -(π / 6) b. r = (4 / (2cosθ - 3sinθ))

1. anonymous

@dan815 @freckles @Luigi0210 @radar @rvc @sleepyjess

2. anonymous

I think for A, r = 2 but I'm not sure

3. freckles

$\theta=\frac{-\pi}{6}$ actually describes a line |dw:1433651567694:dw|

4. freckles

so you know your line will be of the form y=mx

5. freckles

now the job is to find m

6. anonymous

Okay...

7. freckles

$\text{ recall } \frac{y}{x}=\tan(\theta) \\$

8. freckles

plug in your value of theta

9. freckles

you can also write that as y=tan(theta)*x

10. anonymous

I am so confused...

11. anonymous

I thought we were finding m? Are we going backwards now?

12. freckles

m is just a constant value I showed what m was by replacing m with tan(theta) and I got m was tan(theta) by using the equation y/x=tan(theta)

13. anonymous

ohhh, so how do I find out what x and y are since I already know theta?

14. freckles

I knew my equation would look like this because I knew theta=a number is a line $y=mx$

15. freckles

y and x are variables

16. freckles

they vary

17. freckles

you only need to find m aka tan(theta)

18. anonymous

But I already know what theta is for a. I need to know how to find r, right?

19. anonymous

@amilapsn

20. anonymous

@UsukiDoll

21. amilapsn

:)

22. anonymous

Can you help me?

23. amilapsn

You don't need to find r

24. anonymous

Then how am I supposed to graph it, if I don't know what r is?

25. amilapsn

You just need to find the Cartesian form of the polar equation $$\theta = -\frac{\pi}{6}$$

26. anonymous

Okay...how do I do that?

27. anonymous

Do I use tan(theta)

28. anonymous

or arctan (theta)?

29. amilapsn

You know that's a straight line, right?

30. anonymous

yes

31. amilapsn

Do you know how to get a equation of a line?

32. anonymous

no

33. amilapsn

you know the Cartesian form of the line is y=mx+c, right?

34. anonymous

okay, so how would I write that with -pi/6??

35. anonymous

hello...??? @amilapsn

36. amilapsn

Calm down... @Shorty1234 take a deep breath.... :)

37. amilapsn

yes of course.... You have to find m and c... what would be c, (it's easy)?

38. anonymous

-pi/6?

39. amilapsn

No... I'll ask again... you know the Cartesian form of the line is y=mx+c, right? What would m and c represent?

40. amilapsn

(Their meaning)

41. anonymous

m is the slop and c is the y-intercept

42. amilapsn

yes that's it what's the y intercept =c then?

43. anonymous

-pi/6

44. amilapsn

What do you mean by the y intercept?

45. anonymous

is that not what "c" represents ?

46. amilapsn

Yes, it is ....

47. anonymous

okay...so whats your point

48. amilapsn

y- intercept is the y coordinate where straight line cuts the y axis, right?

49. anonymous

yes

50. freckles

sorry lost internet connection do you remember the drawing I made (since you guys are talking about the y-intercept)|dw:1433653769876:dw| where does that cross the y-axis?

51. amilapsn

so?

52. anonymous

it doesnt...so is it zero?

53. amilapsn

:) @freckles

54. amilapsn

you got it..! :)

55. freckles

well it does at 0 (0 is a number :)) so that is how I got y=mx earlier

56. amilapsn

Now we need to find m right?

57. anonymous

so my answer for a would be (0, -pi/6)?

58. amilapsn

@Shorty1234

59. freckles

anyways I will let @amilapsn continue since I was gone so long

60. amilapsn

No....

61. anonymous

.....is there more stuff I have to do??

62. amilapsn

Yes you do...

63. amilapsn

you do have to find m, right?

64. anonymous

ok

65. amilapsn

How do you find m?

66. amilapsn

can you tell me the relation ship between - pi/6 and slope?

67. anonymous

y2 - y1 / x2 - x1

68. amilapsn

That's it! So m would be..............?

69. anonymous

0?

70. freckles

|dw:1433654449002:dw| I'm sorry I said I was going to stay away :p but can you replace -pi/6 and x and y using any trig ratio?

71. freckles

by the way that is a right triangle I made there

72. amilapsn

no.... Can you tell me the relationship between m and pi/6

73. freckles

relate *not replace

74. anonymous

I give up, this makes zero sense

75. freckles

do you know tan(theta)=opp/adj?

76. anonymous

yes

77. freckles

|dw:1433654806812:dw| so tan(-pi/6)=?

78. amilapsn

......in terms of y and x?

79. freckles

yes using the right triangle above

80. anonymous

I got -0.577350269

81. amilapsn

can you express the same in terms of x and y ?

82. anonymous

I don't understand???

83. amilapsn

using the right triangle drawn by freckles...

84. anonymous

but i don't know what x is?

85. amilapsn

yes you don't indeed... use just x and y...

86. anonymous

so express x and y using x and y? that makes no sense

87. freckles

no express tan(-pi/6) in terms of x and y

88. freckles

using the right triangle

89. freckles

I have even labeled the opposite side with the correct measurement and the adjacent side too for the point (x,y)

90. freckles

|dw:1433655304381:dw|

91. anonymous

92. freckles

where opp is y and adj is x right?

93. freckles

so can you say what tan(-pi/6)=? in terms of x and y now?

94. anonymous

tan (-pi/6) = y ???

95. freckles

almost

96. freckles

you forgot the x on denominator

97. anonymous

so tan(-pi/6) / x = y?

98. freckles

:( tan(theta)=y/x you know what theta is just plug in your theta

99. freckles

you are making this way complication your Cartesian equation is suppose to be in terms of y and x you know theta=-pi/6 so you will have an equation in terms of x and y if you replace the theta with -pi/6

100. freckles

lol i'm having trouble with english tonight :p

101. anonymous

.....this makes no sense

102. freckles

why ? I thought the visual made it pretty clear

103. freckles

|dw:1433655836599:dw| if this point is (x,y) then ...

104. freckles

|dw:1433655860761:dw|

105. freckles

106. anonymous

yeah, I dont get this...

107. freckles

let me just do an example I guess say we have $\theta=\frac{\pi}{4}$ again this is a line going through the origin this is of the form y=mx we only need to determine m we can draw this line |dw:1433655974295:dw|

108. freckles

let's say point (x,y) is somewhere on there randomly|dw:1433656006691:dw| just choose a spot

109. freckles

|dw:1433656020961:dw|

110. freckles

well looking at this triangle I can make the observation that $\tan(\frac{\pi}{4})=\frac{y}{x}$ I can be done here

111. freckles

or I can simplify

112. freckles

tan(pi/4)=1

113. freckles

$1=\frac{y}{x} \\ \text{ I can also multiply } x \text{ on both sides } y=x$

114. freckles

y=x is the Cartesian (aka rectangular form ) of the polar equation theta=pi/4

115. freckles

say we have theta=-pi/4 well that would just be tan(-pi/4)=y/x which is -1=y/x which is-x=y

116. freckles

say we have theta=pi/3 that would just be tan(pi/3)=y/x or y=tan(pi/3)x we can evaluate tan(pi/3) $\tan(\frac{\pi}{3})=\frac{\sqrt{3}}{1}=\sqrt{3}$ so theta=pi/3 can be written as the Cartesian equation $y=\sqrt{3}x$

117. anonymous

So my answer would be y = tan(-pi/6)x ?

118. freckles

yes and simplify the tan(-pi/6)

119. anonymous

right,I'll do that. So what about the second problem. Hopefully it what take as long to figure out....

120. freckles

b. r = (4 / (2cosθ - 3sinθ)) multiply both sides by the denominator on right hand side you will need the other two equations to help you to convert this in terms of x and y that you will need rcos(theta)=x and rsin(theta)=y

121. anonymous

ok, thanks

122. freckles

by the way these equations are useful when converting from one to the other $r^2=x^2+y^2 \\ \tan(\theta)=\frac{y}{x} \\ x=r \cos(\theta) \\ y=r \sin(\theta)$ these are your tools to work these particular puzzles

123. freckles

though it is kinda useful knowing why they are and that is why I drew the picture above because it doesn't appear you know why these equations are

124. freckles

|dw:1433657166077:dw| all of those equations can be found just by using this picture

125. amilapsn

|dw:1433657937284:dw|

126. amilapsn

Just a hint: )

127. amilapsn

Things may seem complicated... But they aren't..! Math is the way to sort things out!! :)

128. amilapsn

Hope is the only thing you should have!!

129. amilapsn

Don't give up.. Just try....!

130. amilapsn

You'll find the mathematician inside you..

131. amilapsn

|dw:1433658575750:dw|