Question

Find the points of intersection of the polar graphs. . \(r=2 \cos \theta\) and \(r=2 \sin \theta\) on \([0, \pi]\)

Short answer

The points of intersection are \((0,0)\) and \(\left(1,1\right)\).

Step-by-step solution

Convert Polar to Cartesian

For the equation \(r = 2 \cos \theta\), convert to Cartesian coordinates by multiplying through by \(r\) to get \(r^2 = 2r \cos \theta\), or \(x^2 + y^2 = 2x\). Simplify by completing the square to get \((x-1)^2 + y^2 = 1\).For the equation \(r = 2 \sin \theta\), convert to Cartesian by multiplying through by \(r\), resulting in \(r^2 = 2r \sin \theta\), or \(x^2 + y^2 = 2y\). Simplify by completing the square to get \(x^2 + (y-1)^2 = 1\).

Key concepts

Cartesian Coordinates

Polar coordinates are commonly used in trigonometry and calculus, but sometimes converting to Cartesian coordinates makes problems easier to solve. In the given exercise, we need to switch from polar to Cartesian to identify the points of intersection of the graphs described by the equations \(r = 2\cos\theta\) and \(r = 2\sin\theta\). This involves understanding the relationship between polar and Cartesian systems.The basic conversion formulas are:

• \(x = r\cos\theta\)
• \(y = r\sin\theta\)
• \(r = \sqrt{x^2 + y^2}\)
\ul>Using these, we transform the polar equation \(r = 2\cos\theta\) into Cartesian by multiplying both sides by \(r\), resulting in \(x^2 + y^2 = 2x\). Similarly, for \(r = 2\sin\theta\), multiplying through by \(r\) gives \(x^2 + y^2 = 2y\). These conversions are crucial as they allow us to use algebraic techniques that are easier to apply in Cartesian form.

Completing the Square

After converting the polar equations to Cartesian form, the next step is completing the square. This technique is a method used to transform quadratic equations into a form that’s easier to work with, particularly when identifying circles or other conic sections.For the equation \(x^2 + y^2 = 2x\), we complete the square for the \(x\)-terms. By adding and subtracting 1, we obtain \((x-1)^2 + y^2 = 1\). This shows that the equation describes a circle with center at \((1,0)\) and radius 1. Similarly, for \(x^2 + y^2 = 2y\), we complete the square on the \(y\)-terms by adding and subtracting 1, which results in \(x^2 + (y-1)^2 = 1\). This represents a circle centered at \((0,1)\) with radius 1. Completing the square clarifies the geometric nature of these equations, allowing us to visualize them as circles on the Cartesian plane.

Intersection of Graphs

Finding the intersection of graphs involves determining points where the equations or shapes overlap. In this problem, after we've converted the polar forms to Cartesian and completed the square, we have two circles: one centered at \((1,0)\) and another at \((0,1)\), both with radius 1.To find their intersection, we look for points that satisfy both circle equations:

• \((x-1)^2 + y^2 = 1\)
• \(x^2 + (y-1)^2 = 1\)

Solving these simultaneously involves equating and simplifying both equations. Often, a substitution or elimination method works best. For these particular circles, computational solutions reveal that they intersect at points that lie exactly where the circles touch or overlap, which is more geometric and visual in nature.In this scenario, the intersection points in terms of polar coordinates previously found provide specific angles where both shapes meet on the unit circle between angles 0 and \(\pi\). Translating between coordinate systems allows deeper insights into where these graphs overlap and how they relate in both polar and Cartesian contexts.