r=8cos(theta)+5sin(theta), convert the polar equation into a rectangular equation and then complete the square and determine the center radius...?

Help?! I missed the section, and am not getting anywhere with the text...

In your text, you should find a diagram similar to this one

http://en.wikipedia.org/wiki/File:Polar_to_cartesian.svg

If you label the endpoint of the rotating arm as either
(x,y) in rectangular and (r,Ø) in polar, then

x = rcosØ or cosØ = x/r
y = rsinØ or sinØ = y/r

also x^2 + y^2 = r^2, and tanØ = y/x

then in your equation r = 8(x/r) + 5(y/r)
multiply by r
r^2 = 8x + 5y
x^2 + y^2 = 8x + 5y
x^2 - 8x + y^2 - 5y = 0

This is the equation of a circle. I will assume you know how to complete the square and thus find the centre and radius.

To convert the polar equation r = 8cos(theta) + 5sin(theta) into a rectangular equation, we can use the following relationships:

x = r * cos(theta)
y = r * sin(theta)

Substituting these values into the given polar equation, we get:

x = (8cos(theta) + 5sin(theta)) * cos(theta)
y = (8cos(theta) + 5sin(theta)) * sin(theta)

Now, we can simplify these equations:

x = 8cos^2(theta) + 5sin(theta) * cos(theta)
y = 8cos(theta) * sin(theta) + 5sin^2(theta)

Next, we can simplify further using the trigonometric identities:

cos^2(theta) = (1 + cos(2 theta))/2
sin^2(theta) = (1 - cos(2 theta))/2
sin(theta) * cos(theta) = (sin(2 theta))/2

Applying these identities, we get:

x = 8(1 + cos(2 theta))/2 + 5(sin(2 theta))/2
y = 8(sin(2 theta))/2 + 5(1 - cos(2 theta))/2

Now, we can simplify even further:

x = 4 + 4cos(2 theta) + 5sin(2 theta)
y = 4sin(2 theta) + 5 - 5cos(2 theta)

To complete the square, we can manipulate the equations as follows:

x - 4 = 4cos(2 theta) + 5sin(2 theta)
y - 5 = 4sin(2 theta) - 5cos(2 theta)

Now, we square these equations and use the trigonometric identity:

sin^2(theta) + cos^2(theta) = 1

(x - 4)^2 = (4cos(2 theta) + 5sin(2 theta))^2
(y - 5)^2 = (4sin(2 theta) - 5cos(2 theta))^2

Expanding the equations, we have:

(x - 4)^2 = 16cos^2(2 theta) + 40cos(2 theta)sin(2 theta) + 25sin^2(2 theta)
(y - 5)^2 = 16sin^2(2 theta) - 40cos(2 theta)sin(2 theta) + 25cos^2(2 theta)

Now, we can combine the terms:

(x - 4)^2 + (y - 5)^2 = 16(cos^2(2 theta) + sin^2(2 theta)) + 25(cos^2(2 theta) + sin^2(2 theta))
(x - 4)^2 + (y - 5)^2 = 41(cos^2(2 theta) + sin^2(2 theta))

Using the identity sin^2(theta) + cos^2(theta) = 1, we simplify further:

(x - 4)^2 + (y - 5)^2 = 41

So, the rectangular equation of the given polar equation is (x - 4)^2 + (y - 5)^2 = 41.

Now, we can identify the center and radius of the equation. Comparing the given equation with the standard form equation for a circle:

(x - h)^2 + (y - k)^2 = r^2

We can see that the center of the circle is (4, 5) and the radius is the square root of 41. Therefore, the center of the circle is (4, 5) and the radius is approximately 6.403.

To convert the polar equation r = 8cos(theta) + 5sin(theta) into a rectangular equation, we can use the following relationships:

x = r * cos(theta)
y = r * sin(theta)

Substituting these equations into the given polar equation, we have:

x = (8cos(theta) + 5sin(theta)) * cos(theta)
y = (8cos(theta) + 5sin(theta)) * sin(theta)

Expanding and simplifying these equations, we get:

x = 8cos^2(theta) + 5cos(theta)sin(theta)
y = 8cos(theta)sin(theta) + 5sin^2(theta)

Now, let's simplify these equations further:

x = 8(cos^2(theta) + sin^2(theta)) + 5cos(theta)sin(theta)
x = 8 + 5cos(theta)sin(theta)

y = 5(cos^2(theta) + sin^2(theta)) + 8cos(theta)sin(theta)
y = 8 + 5cos(theta)sin(theta)

Since the terms involving cos(theta)sin(theta) cancel out, we are left with:

x = 8
y = 8

The rectangular equation is x = 8 and y = 8.

To complete the square and determine the center and radius, we need to rewrite the equation in standard form (x - h)^2 + (y - k)^2 = r^2.

In this case, since the coefficients of x and y are both zero, the equation represents a point, not a circle. The point is located at the coordinates (8, 8), and the radius is zero.

Therefore, the center of the point is (8, 8) and the radius is zero.