in the equation r= 5/(sin theta +2cos theta) the letters r and theta represent polar coordinates. Write the equivalent equation using rectangular coordinates.

Thanks :)

recall:

sinØ = y/r and cosØ = x/r

so:
r= 5/(sin theta +2cos theta)
r = 5/(y/r + 2x/r)
r = 5/((2x+ y)/r)
r = 5r/(2x + y)
1 = 5/(2x+y)
2x + y = 5 <----- a straight line

proof:
http://www.wolframalpha.com/input/?i=polar+plot+r%3D+5%2F(sin+theta+%2B2cos+theta)

Why did the polar coordinate go to the therapist?

Because it was having trouble getting along with its imaginary friends!

But to answer your question, to convert from polar coordinates (r, θ) to rectangular coordinates (x, y), you can use the following formulas:

x = r * cos(θ)
y = r * sin(θ)

So, using these formulas, the equivalent equation in rectangular coordinates would be:

x = 5 / (sin(θ) + 2cos(θ))
y = 5 / (sin(θ) + 2cos(θ))

Hope that brings a smile to your face!

To convert the equation from polar coordinates to rectangular coordinates, we can use the following relationships:

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

Substituting these into the given equation, we get:

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

Thus, the equivalent equation in rectangular coordinates is:

(x, y) = (5 * cos(theta))/(sin(theta) + 2cos(theta)), (5 * sin(theta))/(sin(theta) + 2cos(theta))

To write the equivalent equation using rectangular coordinates, we need to express the given equation using x and y.

First, let's recall the conversion formulas between polar and rectangular coordinates:

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

Substituting these values into the given equation, we have:

r = 5 / (sin(theta) + 2 * cos(theta))

x = r * cos(theta)
x = (5 / (sin(theta) + 2 * cos(theta))) * cos(theta)
x = 5 * cos(theta) / (sin(theta) + 2 * cos(theta))

y = r * sin(theta)
y = (5 / (sin(theta) + 2 * cos(theta))) * sin(theta)
y = 5 * sin(theta) / (sin(theta) + 2 * cos(theta))

Thus, the equivalent equation using rectangular coordinates is:

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

Hope this helps!