SOLUTION: convert the rectangular equation y=5x+4 to a polar equation...thanks

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Question 70073: convert the rectangular equation y=5x+4 to a polar equation...thanks
Found 2 solutions by Edwin McCravy, bucky:
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
Convert the rectangular equation y = 5x + 4 to a 
polar equation...thanks

Since sinq = y/r, then y = r·sinq
Since cosq = x/r, then x = r·cosq

So substitute r·sinq for y and r·cosq for x

y = 5x + 4

and get

r·sinq = 5r·cosq + 4

Now solve for r, Get all terms that contain r on the
same side of the equation:

r·sinq - 5r·cosq = 4

Factor out r

r(sinq - 5·cosq) = 4

Divide both sides by the parentheses:

           4
r = ----------------
     sinq - 5·cosq
 
Edwin
Answer by bucky(2189)   (Show Source): You can put this solution on YOUR website!
y = 5x + 4
Let @ represent the angle theta (theta is a letter in the Greek alphabet. Letters in
the Greek alphabet are used in many math books to represent angles just as letters
in our alphabet such as x and y are often used to represent distances. Don't let this
confuse you and don't worry about it. If you don't like the Greek letter theta, just replace
it with a capital letter from our alphabet ... but to avoid confusion, don't use R, X, or Y.
Use something such as A, B, C ...)
Perform the following steps:
(1) for y in the equation substitute r*sin@
.
r represents the magnitude or length in polar form. It originates from the origin and extends to a point on the coordinate system.
.
(2) for x in the equation substitute r*cos@
.
(3) solve for r
Let's do it.
The given equation is y = 5x + 4
Do the substitution of step 1 to get:
.
r*sin@ = 5x + 4
.
Next do the substitution of step 2 to get:
.
r*sin@ = 5r*cos@ + 4
.
Now solve for r. First subtract 5r*cos@ from both sides. This results in:
.
r*sin@ - 5r*cos@ = 4
.
Factor out the r that is common to both terms:
.
r(sin@ - 5cos@) = 4
.
That's a suitable form of the polar equation that is equivalent to y = 5x + 4.
.
Just for grins, let's try a few points to see if they are equal in both the rectangular
and polar forms. Let's try letting the angle @ be 90 degrees -- or if you prefer pi/2
radians. Plug this value of 2 into the polar form and you get:
r[sin(90) - 5cos(90)] = 4
Now recognize that sin(90) = 1 and cos(90) = 0. Substitute these values to get:
r(1 - 5*0) = 4
which simplifies to:
r(1 - 0) = 4 or simply r = 4 with r starting at the origin (0,0).
Therefore, our polar answer is 4/90 which is magnitude 4 at an angle of 90 degrees.
But think about this. Since we measure angles counter-clockwise relative to the x-axis in
quadrant I, the angle of 90 degrees puts us on the positive y-axis. And since we found
that the magnitude of r was 4, the point determined by the polar form 4/90 is 4 units up
the y-axis from the origin. In rectangular form that point would be (0,4). But does
(0,+4) satisfy the equation we were given? Let's see ... Start with
y = 5x + 4
and substitute 0 for x and 4 for y. When you do you get:
4 = 5*0 + 4
which reduces to 4 = 4. It works! Therefore, we have shown that at least for this
example, the polar form r(sin@ - 5cos@) = 4 and the rectangular form y = 5x + 4 produce
the equal answers of 4/90 and (0,4). Let's try letting @ = 0 degrees in the polar form,
solving for r, then translating to an (x,y) answer and see if it solves our rectangular
form. Something interesting and educational happens here. Begin by substituting
0 degrees into the polar form:
r(sin(0) - 5cos(0)) = 4
Recognize that sin(0) = 0 and cos(0) = 1. Substitute to get:
r(0 - 5*1) = 4
Multiply out and you get -5r = 4 and when you divide both sides by -5 you get
r = -5/4
What does it mean when you get a negative value for r. It means that it actually
runs in the opposite direction (180 degrees away) from what you thought it was. In this
case we thought that r would be at 0 degrees, and the negative sign on the magnitude
tells us that the magnitude is positive, but the angle is actually 180 degrees.
.
Using the convention of angle measurement we now know that r = +4/5 but the 180 degree angle
puts it on the negative x-axis. But in rectangular form, a point on the negative
x-axis has a y value of of zero. Therefore in rectangular form we are at the point
(-4/5, 0). Use these values (x = -4/5 and y = 0) in the equation y = 5x + 4 and see if
they don't satisfy the equation. [Hint: they do!]
By now you should feel pretty comfortable in saying that since the polar and rectangular
equations have produced the same points for some easy comparisons, they are probably
equivalent equations and our polar form is likely to be correct.
Hope this hasn't confused you too much, but if you think about it, maybe you'll start getting
the conversion from rectangular to polar form.

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