Converting Equations from Polar to Rectangular
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Key Questions

Answer:
As discussed below.
Explanation:
Polar coordinates are a complementary system to Cartesian coordinates, which are located by moving across an xaxis and up and down the yaxis in a rectangular fashion. While Cartesian coordinates are written as (x,y), polar coordinates are written as (r,Î¸).
In polar coordinates, a point in the plane is determined by its distance r from the origin and the angle theta (in radians) between the line from the origin to the point and the xaxis
The polar coordinate system is a twodimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction
Why is it called Polar form?
In polar coordinates the origin is often called the pole. Because we aren't actually moving away from the origin/pole we know that . However, we can still rotate around the system by any angle we want and so the coordinates of the origin/pole are .Who invented polar coordinates?
The polar coordinate system is an adaptation of the twodimensional coordinate system invented in 1637 by French mathematician RenÃ© Descartes (1596â€“1650). Several decades after Descartes published his two dimensional coordinate system, Sir Isaac Newton (1640â€“1727) developed ten different coordinate systems.Is it helpful?

Answer:
See explanation
Explanation:
Imagine an arm where one end is pinned at the centre of a circular style graph and at the other end is a pencil. Also, the length of this arm can change such that its length is governed by a formula involving the angle of rotation.
Suppose the length of the arm is
#r# and we had one equation that was#r=5sintheta# and a second where#r=5sin(3theta)# Then we would have:
If you so chose you could graph it on standard graph paper. In which case it would look like:

To convert an equation given in polar form (in the variables
#r# and#theta# ) into rectangular form (in#x# and#y# ) you use the transformation relationships between the two sets of coordinates:
#x=r*cos(theta)#
#y=r*sin(theta)#
You have to remember that your equation may need some algebraic/trigonometric manipulations before being transformed into rectangular form; for example, consider:#r[2sin(theta)+3cos(theta)]=2#
#2rsin(theta)+3rcos(theta)=2# Now you use the above transformations, and get:
#2y+3x=2#
Which is the equation of a straight line!A more complicated situation can be the following example:
#theta+pi/4=0#
You can write:
#theta=pi/4#
Take the tangent of both sides and multiply and divide by#r# the left side:
#r/r*tan(theta)=tan(pi/4)#
#(rsin(theta))/(rcos(theta))=1#
Transforming you get:
#y/x=1#
#y=x# 
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