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. $$r=\frac{1}{4}-\cos 4 \theta$$

Short answer

Question: Find the polar coordinates of the points marked on the Cartesian graph and plot them on the polar graph of \(r=\frac{1}{4}-\cos 4\theta\). Answer: To find the polar coordinates of the marked points on the Cartesian graph, first identify their Cartesian coordinates, then use the conversion formulas to find the polar coordinates, and finally plot the points on the polar graph of \(r=\frac{1}{4}-\cos 4\theta\).

Step-by-step solution

Identify the points on the Cartesian graph

Observe the Cartesian graph and find the coordinates of the marked points. Write these points down for further conversion to polar coordinates.

Convert the Cartesian points to polar coordinates

Use the following formula to convert each point from Cartesian coordinates \((x,y)\) to polar coordinates \((r,\theta)\): \(r=\sqrt{x^2+y^2}\) \(\theta=\tan^{-1}\frac{y}{x}\) Remember to find the correct quadrant for the angle based on the x and y values of each point.

Locate the points on the polar graph

After converting the points from Cartesian to polar coordinates, plot them on the given polar graph of \(r=f(\theta)\). Ensure that the distances and angles are accurate when plotting the points.

Key concepts

Cartesian Graph

In mathematics, a Cartesian graph represents data using a two-dimensional coordinate system. This is the standard graph with an x-axis (horizontal) and a y-axis (vertical). Each point on the Cartesian plane is identified by a pair of numbers \(x,y\). The x-coordinate measures the horizontal distance from the origin, while the y-coordinate measures the vertical distance. These coordinates can be used to represent different forms of equations and relationships.
The straight lines, curves, and other shapes plotted on the Cartesian graph help visualize equations, such as the given function \(r=\frac{1}{4}-\cos 4 \theta\). This can greatly assist in analyzing functions and identifying patterns or specific points. The Cartesian system is foundational in many mathematical fields, serving as the basis for more advanced graphing techniques.

Conversion Formulas

To understand the relationship between Cartesian and polar graphs, conversion formulas are crucial. Converting Cartesian coordinates to polar coordinates involves a transformation process using two main formulas.

• The first formula calculates the radial distance \(r\) from the origin: \(r=\sqrt{x^2+y^2}\).
• The second formula determines the angle \(\theta\): \(\theta=\tan^{-1}\frac{y}{x}\).

These formulas are essential because they allow us to switch from one system of representation to another, enabling a broader analysis. However, it's important to be careful when determining \(\theta\). Depending on the quadrant in which the point lies, the angle might need adjustment to ensure it's correct within the polar graph context.

Distance Calculation

Distance calculation in a graph is the process of determining how far one point is from another. In the given task, knowing how to calculate the distance from the origin to each point is essential when converting to polar coordinates. For this, the formula \(r=\sqrt{x^2+y^2}\) is key. This formula calculates the Euclidean distance between the origin (0, 0) and the point \(x,y\) on the Cartesian plane.
Being able to accurately calculate distances ensures that the corresponding points on the polar graph maintain their true positioning. This concept is not only important in graph transformations but also widely used in physics, engineering, and computer science for spatial analysis and modeling.

Trigonometry

Trigonometry plays a significant role in the conversion of Cartesian coordinates to polar coordinates. The trigonometric function \(\tan^{-1}\), which is the inverse of tangent, is used to calculate the angle \(\theta\) when moving from Cartesian to polar coordinates. This angle represents the rotation from the positive x-axis to the line segment joining the origin with our given point. Another trigonometric function, cosine, is part of the given equation \(r=\frac{1}{4}-\cos 4 \theta\). Here, cosine's repetitive nature helps form patterns or shapes, such as circles or waves, on a graph, which can be effectively represented in a polar plot. Understanding the fundamentals of these trigonometric functions allows better visualization and analysis of mathematical relationships in both Cartesian and polar systems. Trigonometry is a powerful tool that allows us to bridge different mathematical realms and solve complex problems efficiently.