How do you convert $r = 2 tan θ sec θ$ $r = 2tanthetasectheta$ into cartesian form?

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How do you convert $r = 2 tan θ sec θ$ $r = 2tanthetasectheta$ into cartesian form?

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Answer 1 · 2016-02-10T22:19:27

By substitution $y = \frac{1}{2} \cdot x^{2}$ $y = 1/2*x^2$

Explanation:

First convert the expression into terms of $cos (θ)$ $cos(theta)$ and $sin (θ)$ $sin(theta)$
$r = 2 \cdot \frac{sin (θ)}{cos (θ)} \cdot \frac{1}{cos (θ)}$ $r = 2*sin(theta)/cos(theta) *1/cos(theta)$
$r {cos}^{2} (θ) = 2 \cdot sin (θ)$ $rcos^2(theta) = 2*sin(theta)$
Now the normal substitutions are $x = r cos (θ)$ $x=rcos(theta)$ and $y = r sin (θ)$ $y=rsin(theta)$
$:. sin(theta) = y/r$ and $cos(theta) = x/r$

Substituting these values into the expression above gives
$r*(x/r)^2 = 2*y/r$
$cancel(r)*x^2/r^cancel(2) = 2*y/r$
$x^2 = r*2*y/r =2* y$
$:.y =1/2 x^2$

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How do you convert $r = 2 tan θ sec θ$ $r = 2tanthetasectheta$ into cartesian form?

1 Answer

Explanation:

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How do you convert $r = 2 tan θ sec θ$ $r = 2tanthetasectheta$ into cartesian form?