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Given that x=1+2tan theta and y=3sec theta - 4, by eliminating theta, determine a relationship between x and y only !


Given that x=sin theta and y=cos^2theta - 3, by eliminating theta determine a relationship between x and y only

Respuesta :

arthurpdc
[tex]
x=1+2\tan\theta\\\\
2\tan\theta=x-1\\\\
\tan\theta=\frac{1}{2}(x-1)\\\\\\
y=3\sec\theta-4\\\\
3\sec\theta=y+4\\\\
\sec\theta=\frac{1}{3}(y+4)[/tex]

We'll use the formula below

[tex]
\tan^2\alpha+1=\sec^2\alpha\iff \sec^2\alpha-\tan^2\alpha=1[/tex]

So:

[tex]
\sec^2\theta-\tan^2\theta=1\\\\
(\frac{1}{3}(y+4))^2-(\frac{1}{2}(x-1))^2=1\\\\
\boxed{\dfrac{(y+4)^2}{9}-\dfrac{(x-1)^2}{4}=1}
[/tex]

-----------------//----------------

[tex]
x=\sin\theta\\\\\\
y=\cos^2\theta-3\\\\
\cos^2\theta=y+3[/tex]

We'll use the formula below

[tex]
\sin^2\alpha+\cos^2\alpha=1[/tex]

So:

[tex]
\sin^2\theta+\cos^2\theta=1\\\\
x^2+(y+3)=1\\\\
\boxed{x^2+y=-2}
[/tex]

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