<PRE><B>Question 70001</B>
the rectangular coordinates of a particular point are x=7, y=-24, find the polar coordinates...

Note that since the value of y is negative and the value of x is positive.  This puts
the point in quadrant IV.

The magnitude of the radius of is the hypotenuse of a right triangle with legs of length 
7 and 24.  Use the Pythagorean theorem to find this hypotenuse (H).

{{{H = sqrt(7^2 + 24^2)}}}

{{{H = sqrt(49+576) = sqrt(625) = 25}}}

That gives us Magnitude.  All we need now is the angle.  The ratio of the angle is legs
of the triangle are: 
{{{y/x=(-24/7)= -3.428571429}}}

This meets the definition of the tangent of the angle whose opposite side is -24 and whose
adjacent side is +7.  Notice how this falls into quadrant IV = Since this ratio is the 
tangent, by applying arctangent on a calculator, we find that the angle is -73.73979 degrees.  

The polar form can be written in two ways:

   25/-73.73979

which means a magnitude of 25 units with an angle found by rotating the magnitude
clockwise from positive portion of the x-axis by the amount 73.73979 degrees.

The angle can also be established by rotating (360-73.73979 or 286.26021 degrees)
counter-clockwise from the positive portion of the x-axis.  This would be written as:

    25/286.26021

I hope this helps you to understand the relationship between the Cartesian and polar representations
of points and specifically how to change from rectangular to polar form.

You may also find it helpful to make a rough sketch of the rectangular point (7,-24) and
draw the magnitude (radius) from the origin to this point.  Measure out 7 on the positive
x-axis and the vertical distance from the point (7,0) to the point (7, -24).  This will
help you to see the right triangle (0,0) to (7,0), from (7,0) to (7, -24), and then from
(7, -24) to (0,0).  The two points (7, -24) to (0,0) are the ends of the radius and the
Magnitude of this line can also be found using the formula for the distance between two points.
the tangent of the angle is the ratio of the length of the line (7,0) to (7, -24) over 
the length of the line from (0,0) to (7,0). Such a rough sketch will help you to visualize
the problem.

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