Eliminating parameters

[hgaon001]

I have absolutely no idea how to do this problem.

Eliminate the parameter to find the rectangular equation of the curve (there should be no trig functions or inverse trig functions in your final answer)

x=2cost, y=3(sec(t)^2)

im guessing you have to do the dy/dx? but i dont understand how it would get rid of the trig functions...

 

[galactus]

Solve \(\displaystyle x=2cos(t)\) for t and sub into \(\displaystyle y=3sec^{2}(t)\)

\(\displaystyle t=cos^{-1}(\frac{x}{2})\)

Sub into y.

\(\displaystyle 3sec^{2}(cos^{-1}(\frac{x}{2}))=\frac{12}{x^{2}}\)

Can you work this out?.

Draw a triangle and label the trig functions.

By the diagram we can see that \(\displaystyle cos(y)=\frac{x}{2}\Rightarrow cos^{-1}(\frac{x}{2})=y\)

By Pythagoras, the side opposite y is equal to \(\displaystyle \sqrt{4-x^{2}}\)....we don't really need this, but I thought I would point it out.

Therefore, \(\displaystyle sec(y)=\frac{2}{x}\Rightarrow sec^{2}(y)=\frac{4}{x^{2}}\)

Sub in y and multiply by 3 and get \(\displaystyle 3sec^{2}(cos^{-1}(\frac{x}{2}))=\frac{12}{x^{2}}\)

 

[Deleted member 4993]

I have absolutely no idea how to do this problem.

Eliminate the parameter to find the rectangular equation of the curve (there should be no trig functions or inverse trig functions in your final answer)

x=2cost, y=3(sec(t)^2)

im guessing you have to do the dy/dx? but i dont understand how it would get rid of the trig functions...

2 cos(t) = x

cos(t) = x/2

sec(t) = 1/cos(t) = 2/x

y = 3 sec[sup:lh0sl9qg]2[/sup:lh0sl9qg](t) = 3 * (2/x)[sup:lh0sl9qg]2[/sup:lh0sl9qg]

Now simplify further....

 

[fasteddie65]

I have absolutely no idea how to do this problem.

Eliminate the parameter to find the rectangular equation of the curve (there should be no trig functions or inverse trig functions in your final answer)

x = 2 cos t, y = 3 (sec t)^2

im guessing you have to do the dy/dx? but i dont understand how it would get rid of the trig functions...

Did you notice that cos and sec are reciprocal functions? So, it cos t = x/2, then sec t = 2/x. So sec^2 t= (2/x)^2 and 3 sec^2 t = 3(2/x)^2 = 3(4/x^2) = 12/x^2.

So y = 12/x^2. This requires a restricted domain, of course.