How to approach this math problem?

Question

I'm working on polar coordinates and I would like to know how to approach this problem. Please do not solve for me since I do not wish to know the answer anyway. Here is the problem for reference:

To the nearest integer, what is the distance between the points of intersection of the polar curves:

r = 4cos(theta) and r = 2 / [cos(theta) + sin(theta)]

Should I convert to x and y form first? And then I don't understand what the distance between the points of interesection means here...should there even be a distance between the "Points of intersection"?

Also, this is to be done without a graphing calculator or utility.

Answer 1

Actually, I would recommend staying in polar form until you've found the solution. Since you have r= something for both equations, you can handle them just like any other equation - that is, substitute 4 cos θ for r to get 4 cos θ = 2/(cos θ + sin θ), then solve for θ. You'll have to compute an inverse trig function at the end - at this point, it will suffice to consider only those values of θ between 0 and 2π, since θ only appears in trig functions in your equations, which are periodic with period 2π (i.e. the point given by θ=π will be the same point as θ=3π - note that if θ appeared outside a trig function in either equation, this would not necessarily be true). When you consider the values of θ in that range, you should get exactly two solutions. Find the r for each of these solutions (using the first equation, since it's simpler), and THEN convert to rectangular form. You will then have two points given by their rectangular representations - these are the points of intersection of your curves. Now just find the distance between them.

Answer 2

First set the two expression equal to each other:4*cos(x)=2/[cos(x+sin(x)]. Simplify this to get:cos^2(x)+cos(x)*sin(x)=1/2. Now, graph both sides of this equation and use the calculator to approximate the value of x--your theta. Once you have more then one set of thetas (corresponding to more then one intersection) calculate the x and y coordinates to the points that they refer to. Finally calculate the distance between these points.

Answer 3

Try to solve the equation when both values of r are equal. This equation will involve both sin and cos, and to solve it you'll need to convert everything to the functions of the double angle. If you need more help, ask.

Answer 4

nicely in certain circumstances if its plausible you could list, draw, use algorithms or maybe round up/down even though it in many cases relies upon on how the question is layed out and what it asks you to do wish it helps