anonymous
  • anonymous
Transform each polar equation to an equation in rectangular coordinates and identify its shape. a. θ = -(π / 6) b. r = (4 / (2cosθ - 3sinθ))
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
@dan815 @freckles @Luigi0210 @radar @rvc @sleepyjess
anonymous
  • anonymous
I think for A, r = 2 but I'm not sure
freckles
  • freckles
\[\theta=\frac{-\pi}{6}\] actually describes a line |dw:1433651567694:dw|

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freckles
  • freckles
so you know your line will be of the form y=mx
freckles
  • freckles
now the job is to find m
anonymous
  • anonymous
Okay...
freckles
  • freckles
\[\text{ recall } \frac{y}{x}=\tan(\theta) \\\]
freckles
  • freckles
plug in your value of theta
freckles
  • freckles
you can also write that as y=tan(theta)*x
anonymous
  • anonymous
I am so confused...
anonymous
  • anonymous
I thought we were finding m? Are we going backwards now?
freckles
  • 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)
anonymous
  • anonymous
ohhh, so how do I find out what x and y are since I already know theta?
freckles
  • freckles
I knew my equation would look like this because I knew theta=a number is a line \[y=mx\]
freckles
  • freckles
y and x are variables
freckles
  • freckles
they vary
freckles
  • freckles
you only need to find m aka tan(theta)
anonymous
  • anonymous
But I already know what theta is for a. I need to know how to find r, right?
anonymous
  • anonymous
@amilapsn
anonymous
  • anonymous
@UsukiDoll
amilapsn
  • amilapsn
:)
anonymous
  • anonymous
Can you help me?
amilapsn
  • amilapsn
You don't need to find r
anonymous
  • anonymous
Then how am I supposed to graph it, if I don't know what r is?
amilapsn
  • amilapsn
You just need to find the Cartesian form of the polar equation \(\theta = -\frac{\pi}{6}\)
anonymous
  • anonymous
Okay...how do I do that?
anonymous
  • anonymous
Do I use tan(theta)
anonymous
  • anonymous
or arctan (theta)?
amilapsn
  • amilapsn
You know that's a straight line, right?
anonymous
  • anonymous
yes
amilapsn
  • amilapsn
Do you know how to get a equation of a line?
anonymous
  • anonymous
no
amilapsn
  • amilapsn
you know the Cartesian form of the line is y=mx+c, right?
anonymous
  • anonymous
okay, so how would I write that with -pi/6??
anonymous
  • anonymous
hello...??? @amilapsn
amilapsn
  • amilapsn
Calm down... @Shorty1234 take a deep breath.... :)
amilapsn
  • amilapsn
yes of course.... You have to find m and c... what would be c, (it's easy)?
anonymous
  • anonymous
-pi/6?
amilapsn
  • 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?
amilapsn
  • amilapsn
(Their meaning)
anonymous
  • anonymous
m is the slop and c is the y-intercept
amilapsn
  • amilapsn
yes that's it what's the y intercept =c then?
anonymous
  • anonymous
-pi/6
amilapsn
  • amilapsn
What do you mean by the y intercept?
anonymous
  • anonymous
is that not what "c" represents ?
amilapsn
  • amilapsn
Yes, it is ....
anonymous
  • anonymous
okay...so whats your point
amilapsn
  • amilapsn
y- intercept is the y coordinate where straight line cuts the y axis, right?
anonymous
  • anonymous
yes
freckles
  • 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?
amilapsn
  • amilapsn
so?
anonymous
  • anonymous
it doesnt...so is it zero?
amilapsn
  • amilapsn
:) @freckles
amilapsn
  • amilapsn
you got it..! :)
freckles
  • freckles
well it does at 0 (0 is a number :)) so that is how I got y=mx earlier
amilapsn
  • amilapsn
Now we need to find m right?
anonymous
  • anonymous
so my answer for a would be (0, -pi/6)?
amilapsn
  • amilapsn
@Shorty1234
freckles
  • freckles
anyways I will let @amilapsn continue since I was gone so long
amilapsn
  • amilapsn
No....
anonymous
  • anonymous
.....is there more stuff I have to do??
amilapsn
  • amilapsn
Yes you do...
amilapsn
  • amilapsn
you do have to find m, right?
anonymous
  • anonymous
ok
amilapsn
  • amilapsn
How do you find m?
amilapsn
  • amilapsn
can you tell me the relation ship between - pi/6 and slope?
anonymous
  • anonymous
y2 - y1 / x2 - x1
amilapsn
  • amilapsn
That's it! So m would be..............?
anonymous
  • anonymous
0?
freckles
  • 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?
freckles
  • freckles
by the way that is a right triangle I made there
amilapsn
  • amilapsn
no.... Can you tell me the relationship between m and pi/6
freckles
  • freckles
relate *not replace
anonymous
  • anonymous
I give up, this makes zero sense
freckles
  • freckles
do you know tan(theta)=opp/adj?
anonymous
  • anonymous
yes
freckles
  • freckles
|dw:1433654806812:dw| so tan(-pi/6)=?
amilapsn
  • amilapsn
......in terms of y and x?
freckles
  • freckles
yes using the right triangle above
anonymous
  • anonymous
I got -0.577350269
amilapsn
  • amilapsn
can you express the same in terms of x and y ?
anonymous
  • anonymous
I don't understand???
amilapsn
  • amilapsn
using the right triangle drawn by freckles...
anonymous
  • anonymous
but i don't know what x is?
amilapsn
  • amilapsn
yes you don't indeed... use just x and y...
anonymous
  • anonymous
so express x and y using x and y? that makes no sense
freckles
  • freckles
no express tan(-pi/6) in terms of x and y
freckles
  • freckles
using the right triangle
freckles
  • freckles
I have even labeled the opposite side with the correct measurement and the adjacent side too for the point (x,y)
freckles
  • freckles
|dw:1433655304381:dw|
anonymous
  • anonymous
so opp/adj ??
freckles
  • freckles
where opp is y and adj is x right?
freckles
  • freckles
so can you say what tan(-pi/6)=? in terms of x and y now?
anonymous
  • anonymous
tan (-pi/6) = y ???
freckles
  • freckles
almost
freckles
  • freckles
you forgot the x on denominator
anonymous
  • anonymous
so tan(-pi/6) / x = y?
freckles
  • freckles
:( tan(theta)=y/x you know what theta is just plug in your theta
freckles
  • 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
freckles
  • freckles
lol i'm having trouble with english tonight :p
anonymous
  • anonymous
.....this makes no sense
freckles
  • freckles
why ? I thought the visual made it pretty clear
freckles
  • freckles
|dw:1433655836599:dw| if this point is (x,y) then ...
freckles
  • freckles
|dw:1433655860761:dw|
freckles
  • freckles
tan(-pi/6)=opp/adj where opp=y and adj=x
anonymous
  • anonymous
yeah, I dont get this...
freckles
  • 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|
freckles
  • freckles
let's say point (x,y) is somewhere on there randomly|dw:1433656006691:dw| just choose a spot
freckles
  • freckles
|dw:1433656020961:dw|
freckles
  • freckles
well looking at this triangle I can make the observation that \[\tan(\frac{\pi}{4})=\frac{y}{x} \] I can be done here
freckles
  • freckles
or I can simplify
freckles
  • freckles
tan(pi/4)=1
freckles
  • freckles
\[1=\frac{y}{x} \\ \text{ I can also multiply } x \text{ on both sides } y=x\]
freckles
  • freckles
y=x is the Cartesian (aka rectangular form ) of the polar equation theta=pi/4
freckles
  • 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
freckles
  • 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 \[y=\sqrt{3}x\]
anonymous
  • anonymous
So my answer would be y = tan(-pi/6)x ?
freckles
  • freckles
yes and simplify the tan(-pi/6)
anonymous
  • anonymous
right,I'll do that. So what about the second problem. Hopefully it what take as long to figure out....
freckles
  • 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
anonymous
  • anonymous
ok, thanks
freckles
  • 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
freckles
  • 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
freckles
  • freckles
|dw:1433657166077:dw| all of those equations can be found just by using this picture
amilapsn
  • amilapsn
|dw:1433657937284:dw|
amilapsn
  • amilapsn
Just a hint: )
amilapsn
  • amilapsn
Things may seem complicated... But they aren't..! Math is the way to sort things out!! :)
amilapsn
  • amilapsn
Hope is the only thing you should have!!
amilapsn
  • amilapsn
Don't give up.. Just try....!
amilapsn
  • amilapsn
You'll find the mathematician inside you..
amilapsn
  • amilapsn
|dw:1433658575750:dw|

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