Question

find cartesian equation of the curve traced out by
x=a cos^3(theta), y=a sin^3(theta)

Answer 1

ok, do u have any idea how to start ? have u tried ?

Answer 2

i would like to give you hint: use
\(\large \sin^2 t+\cos^2t=1\)

Answer 3

I got to cos^2(theta)+sin^2(theta)=x/(acostheta)+y/(asintheta)

Answer 4

thus 1=x/asec(theta)+y/a csc(theta)

Answer 5

basically, u need to eliminate theta from those 2 equations, right ?

Answer 6

yes but I'm at a loss now

Answer 7

thats not cartesiian form...

Answer 8

Once you get rid of theta then it's cartesian right?
I just don't know how to get rid of sec and csc now

Answer 9

right.
ok, let me give you another hint :
x/a = cos^3 theta
so,
(x/a)^(2/3) = cos^2 theta
did u get this ?

Answer 10

ok... but if i substitute that into 1=(x/a)sectheta+(y/a)csctheta, I get 1=cos^3 theta/ cos theta + sin^3 theta/sin theta, so basically 1=cos^2 theta + sin^2 theta, which is 1=1...

Answer 11

don't do that!
i found cos^2 theta =....
can u find sin^2 theta =... ?

Answer 12

do you mean cos^2 theta= x/(a cos theta)

Answer 13

no!
x/a = cos^3 theta
so,
(x/a)^(2/3) = cos^2 theta
did u get this ?

Answer 14

i just raised the power of 2/3 on both sides to convert cos^3 theta to cos^2 theta....

Answer 15

ok but where am I supposed to substitute that in for?

Answer 16

u got cos^2 theta as (x/a)^(2/3) right ?
now can u find, similarly, sin^2 theta ?
in terms of y and a ?

Answer 17

it's (y/a)^(2/3)

Answer 18

correct.
now just add the 2 equations
and you would have successfully eliminated the theta

Answer 19

just because cos^2 theta +sin^2 theta =1

Answer 20

ohhh ok

Answer 21

would the simplest answer then be 1=(x/a)^(2/3)+(y/a)^(2/3)

Answer 22

yes, exactly.

Answer 23

ok got it
THANK YOU SO MUCH!!!!

Answer 24

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