Intersection pts of polar equations

In summary, to find the area of the region between the curves r = sin(θ) and r = sin(2θ), the first step is to find the points of intersection by setting the equations equal to each other and solving for θ. Using the trigonometric identity sin(2x) = 2sin(x)cos(x), we can find that the critical intersection point occurs at θ = π/3. This information can then be used to continue solving the problem.
  • #1
n00neimp0rtnt
15
0

Homework Statement


I have to find the area of the region that lies inside the curves:

r = sin(θ)
r = sin(2θ)


The Attempt at a Solution



I'm assuming the first step would be to find the points of intersection so I know WHERE to integrate from/to, so I set the equations equal to each other:

sin(θ) = sin(2θ)


arcsin both sides:
θ = 2θ

And I'm stuck. Analysis of the graph shows that the most crucial intersection point occurs at or very close to 75º, but I would like to be able to show that.
 
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  • #2
Taking arcsin of both sides will only give you some solutions. Try using sin(2x)=2*sin(x)*cos(x).
 
  • #3
Thanks a lot! In that case...

sin(θ) = 2sin(θ)cos(θ)
1 = 2cos(θ)
cos(θ) = 1/2
θ = π/3

That should help me get the rest of the problem, thanks again! =]
 

Related to Intersection pts of polar equations

1. What are polar equations?

Polar equations are mathematical expressions that represent curves or shapes in two-dimensional space using polar coordinates. These equations use the distance from the origin (r) and the angle from a fixed reference line (θ) to describe a point's position.

2. How do you find the intersection points of polar equations?

To find the intersection points of polar equations, you need to set the two equations equal to each other and solve for the values of r and θ that satisfy both equations. This can be done algebraically or by graphing the equations on a polar coordinate plane.

3. Can polar equations have more than two intersection points?

Yes, polar equations can have multiple intersection points, just like any other type of equation. The number of intersection points depends on the complexity of the equations and the shapes they represent.

4. How do you graph polar equations?

To graph polar equations, you first need to plot points by substituting different values for r and θ into the equation. Then, connect the points to create a smooth curve. It is also helpful to plot the key points of the equation, such as the origin, maximum/minimum points, and the asymptotes.

5. Are there any special cases when finding intersection points of polar equations?

Yes, there are some special cases when finding intersection points of polar equations. These include when one equation is a circle or a line, when the equations have a common factor, or when the equations have a symmetry such as a vertical or horizontal reflection. In these cases, the number of intersection points may be limited or the equations may have no intersection points at all.

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