How do you convert #r = 2 csc(theta) # to rectangular form?

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Answer 1 · 2018-07-11T08:45:25

y = 2

#r >=2, csc theta >=1 #

Conversion formula: #r ( cos theta, sin theta ) = ( x, y ), #

with #r = sqrt ( x^2 + y^2 ) >= 0#

For #theta ne kpi, k = 0, +-1, +-2, +-3, ...,#

Here, #r sin theta = y = 2#. See graph for this straight line y = 2,

parallel to x-axis, at a height 2 units.

graph{y - 2 - 0.000000001 x = 0}

Please note that #theta in ( 0, pi)#.

Some nuances, relating to this polar form:

Of course, the same line would be periodically created, for

#Q_1 and Q_2# # theta in ( 2 k pi, (2 k + 1 ) pi ), k = 0, +-1, +-1, +-3, ...#

# r < 0#, for #Q_3 and Q_4#

#theta in ( (2 k - 1 ) pi, 2 k pi ), k = 0, +-1, +-1, +-3, ...#