c4 - Parametric Equations

dhokes

Could someone help me with the following questions plz:

1. find the Cartesian equation of the curve:
x=2cosecØ
y=2cotØ

2. find the Cartesian equation of the curve:
x=2cosØ-1
y=3+2sinØ

Cheers

Hi there,
You can use trigonometric identities for these.
For example, for the first question you can write cosecØ and cotØ in terms of x and y respectively, then use a standard trigonometric identity you should know that connects cosecØ and cotØ for all values of Ø.
The solution to Q2 is very similar but with a different identity that should be even more familiar to you.

Reply 2

OP

thanks, i've managed to do them now

Reply 3

mysticboy

I'll talk you through the first question.
Right, for parametric questions involving trig functions its always a good idea to look at the sin^2 + cos^2 = 1 formula. If we divide this by sin^2 we get 1 + cot^2 = cosec^2. This is particularly useful to us. We can write x/2 = cosec(theta) and y/2 = cot(theta). Sub these into the equation I've just derived before and we get 1 + (y/2)^2 = (x/2)^2 which is the equation of this curve.
I'll let you work the second one out yourself but you do it in exactly the same way. I was always told to "elimate the variable that leaves you with just x and y in the equation" but I was never sure what that meant with trig. You just look for an identity that links the two x and y equations and generally speaking sin^2 + cos^2 = 1 will do this for you, although I have seen double angle formulae used for this purpose too.
Have a go at the second one yourself and post your answer back and we'll check it for you. If anyone wants to write what I put in tex then feel free, I don't have time to work it out myself now. Thanks.

Reply 4

OP

I'm having trouble with this one as I don't know which trig function to use:

x=sin^2Ø
y=sin^22Ø

mysticboy, cheers, that's exactly what i did.

Reply 5

Do you mean

x=\sin^2 \theta,y=\sin^2 2\theta

?
If so, try writing

\sin^2 2\theta

in terms of single angles.

Reply 6

OP

Yeah I do mean that and does

sin^22Ø = sin^2Ø + sin^2Ø

??

Reply 7

Nope,

\sin^2 2\theta \neq 2\sin^2 \theta

. Indeed,

\sin a\theta \neq a\sin \theta

.
Start from the identity for

\sin 2\theta

in terms of

\sin \theta

and

\cos \theta

.

Reply 8

OP

Gaz031

Nope,

\sin^2 2\theta \neq 2\sin^2 \theta

. Indeed,

\sin a\theta \neq a\sin \theta

.
Start from the identity for

\sin 2\theta

in terms of

\sin \theta

and

\cos \theta

.

sin2Ø = 2sinØcosØ, so:

sin^22Ø = 2sin^2Øcos^2Ø

Reply 9

dhokes

sin2Ø = 2sinØcosØ, so:

sin^22Ø = 2sin^2Øcos^2Ø

Rethink the second line.

Reply 10

Christophicus

18

dhokes

sin2Ø = 2sinØcosØ, so:

sin^22Ø = 2sin^2Øcos^2Ø

sin²2Ø = (2sinØcosØ)² = 4sin²Øcos²Ø

Reply 11

OP

Thanks Windowmaker.

I have another question which is more complex.

Find the area of the region between the following curves, which are defined parametrically, adn the x-axis between the given values of Ø.

Between Ø=π/6 and Ø=π/3
x=2sinØ
y=sin^2Ø

I would be very grateful if someone could plz tell me how to answer the question.

Thanks

Reply 12