If we select the ray l to it is in the hopeful x-axis, climate a allude P in the plane has both cartesian collaborates ( x,y) and also polar works with (r, q).
(2)The meaning of the sine and also cosine features imply that (x,y) is given in regards to ( r,q) by
x = r cos(q) , y = r sin(q)
(1)Solving because that r and q then yields the identities
r2 = x2+y2 and tan( q) = y

EXAMPLE 3 convert the suggest ( 4, p/4) frompolar collaborates into cartesian coordinates, and also then display that (2) counter it earlier into polar .

Solution:To execute so, we let r = 4 and let q = p/4 in (1) come obtain
x = 4cosæè p
4öø = 2Ö2, y = 4sinæè p
4öø = 2Ö2 To map back, we an alert that
r2 = x2 + y2 = 8 + 8 = 16, r = 4
and the y/x = 1 means that tan(q) = 1, q = p/4.

If we substitute x = r cos(q) and also y = r sin(q) right into a curve g(x,y) = k, climate the result
g( rcos(q), rsin(q) ) = k
is referred to as the pullback the the curve right into polar coordinates. The identification r2 = x2+y2 is regularly used in pulling a curve back into polar coordinates.Forexample, x2+y2 = R2 for a continuous R > 0 has actually a pullback the
r2 = R2 Þ r = R
Similarly, currently of the kind y = mx become
r sin(q) = m r cos(q) Þ sin(q) = m cos(q) Þ tan(q) = m
This matches instance 4 in the critical section, in i m sorry we saw that the name: coordinates curves because that the polar name: coordinates transformation
T( r, q ) = á r cos(q), r sin(q)ñ
are circles centered at the origin and also lines with the origin, respectively.
Also, it shows that whenever possible, we have to solve because that r to obtain a duty of the type r = f(q) .EXAMPLE 4 convert the curve x2 + (y - 1)2 = 1 into polarcoordinates, and also then deal with for r, if possible.

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Solution: Expanding leads to x2 + y2 -2y + 1 = 1 , so the To execute so, we change y r2 and also let x = rcos( q) :
r2 - 2r sin(q) + 1 = 1
Solving because that r climate yields
r = 2 sin(q)
the is, r = 2 sin(q) is a circle ofradius 1 centered at ( 1,0) . In general, a curve the the type r = 2acos( q) is a circle of radius | a| focused at ( a,0) and also a curve the the type r = 2asin(q) is a circle of radius | a| focused at (0, a) .