Question

A Cartesian and a polar graph of \(r=f(\theta)\) are given in the figures. Mark the points on the polar graph that correspond to the points shown on the Cartesian graph. $$r=\sin (1+3 \cos \theta)$$

Short answer

Question: Given a set of points on the Cartesian graph, find the corresponding points on the polar graph of the equation \(r = \sin(1 + 3\cos\theta)\). Solution: To find the corresponding points on the polar graph, follow these steps: 1. Convert each Cartesian point (x, y) into polar coordinates (r, θ) using the equations: $$r = \sqrt{x^2 + y^2},$$ $$\theta = \arctan\left(\frac{y}{x}\right).$$ 2. Check if the equation \(r = \sin(1 + 3\cos\theta)\) holds for each obtained polar point (r, θ). 3. If the equation is true for a point (r, θ), it is a corresponding point on the polar graph. Mark this point on the polar graph.

Step-by-step solution

Convert Cartesian coordinates to polar coordinates

To convert a Cartesian point \((x, y)\) into polar coordinates \((r, \theta)\), we use the following equations: $$r = \sqrt{x^2 + y^2},$$ $$\theta = \arctan\left(\frac{y}{x}\right).$$ Apply these equations to each point on the Cartesian graph to determine their polar coordinate representation.

Find the matching points on the polar graph

Now that we have the polar coordinates for each point, we need to locate these points on the polar graph given by the polar equation \(r = \sin(1 + 3\cos\theta)\). 1. For each point \((r, \theta)\) obtained in Step 1, check if the equation \(r = \sin(1 + 3\cos\theta)\) holds. 2. If the equation is true for a point \((r, \theta)\), mark this point on the polar graph. By following these steps, we can find the points on the polar graph that correspond to the points shown on the Cartesian graph.

Key concepts

Cartesian Coordinates

The Cartesian coordinate system is one of the most common ways to describe the location of a point in a plane. It uses two numbers called coordinates, usually represented by \((x, y)\).

• x-coordinate: This value indicates how far left or right a point is from the vertical y-axis.
• y-coordinate: This value shows how far up or down a point is from the horizontal x-axis.

The origin, where the x-axis and y-axis intersect, has the coordinates \((0, 0)\). Understanding Cartesian coordinates is fundamental to graphing and analyzing points on a plane, laying the foundation for more complex geometry, like polar coordinates.

Coordinate Conversion

Converting between Cartesian and polar coordinates involves translating the point's position from one system to another. This is crucial in linking different mathematical perspectives.

• From Cartesian to Polar: Use the formulas \[ r = \sqrt{x^2 + y^2} \] for the radial distance, and \[ \theta = \arctan\left(\frac{y}{x}\right) \] for the angle.
• From Polar to Cartesian: Use \[ x = r \cos(\theta) \] and \[ y = r \sin(\theta) \] if you must reverse the process.

These conversions allow understanding and moving points seamlessly between different types of graphs, making complex problems more manageable.

Polar Graph

A polar graph represents points using polar coordinates \((r, \theta)\). These graphs provide insights into relationships and patterns where circular symmetry is involved.

• Radius (r): The distance from the origin to the point.
• Angle (θ): The direction from the positive x-axis to the point.

The shape and behavior of polar graphs can be vastly different from Cartesian graphs. Polar equations, like \(r = \sin(1 + 3\cos\theta)\), often produce unique and intricate patterns not easily observed in regular Cartesian coordinates.

Equation Verification

Equation verification is the process of determining if a specific point satisfies a given polar equation. In our example, we need to check if the points converted into polar coordinates from Cartesian coordinates fit the equation \(r = \sin(1 + 3\cos\theta)\).

• Substitute the \((r, \theta)\) values into the left-side of the equation.
• Calculate and confirm if it equals the right side.

If the equation holds true for a particular point, it is valid and can be plotted on the polar graph.
This process ensures that each point accurately represents a solution to the polar equation, tying both graphs together.