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)#