How do you convert $x^2 - y^2 = 1$ in polar form?

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Answer 1 · 2015-12-03T23:24:46

Use conversion formulas and algebraic manipulation to find the polar form of
$r = 1/sqrt(cos2theta)$

The question How do you convert rectangular coordinates to polar coordinates? has a list of equations used to convert between rectangular and polar coordinates, along with derivations.

For this problem, we will use
$x = rcos(theta)$
$y = rsin(theta)$

Substituting these into the given equation gives

$(rcos(theta))^2 - (rsin(theta))^2 = 1$

$=> r^2cos^2(theta)-r^2sin^2(theta) = 1$

$=> r^2(cos^2(theta)-sin^2(theta)) = 1$ (trig identity)

$=>r^2cos(2theta) = 1$ (note that from this we know $cos(2theta)>0$ )

$=> r^2 = 1/cos(2theta)$

$=> r = 1/sqrt(cos2theta)$

How do you convert x^2 - y^2 = 1 x^2 - y^2 = 1 in polar form?