Question

Plot the points whose polar coordinates are \(\left(3, \frac{1}{3} \pi\right)\), \(\left(1, \frac{1}{2} \pi\right),\left(4, \frac{1}{3} \pi\right),(0, \pi),(1,4 \pi),\left(3, \frac{11}{7} \pi\right),\left(\frac{5}{3}, \frac{1}{2} \pi\right)\), and \((4,0) .\)

Short answer

Convert polar to Cartesian and plot points on a graph.

Step-by-step solution

Understanding Polar Coordinates

In polar coordinates, each point is described by a radius and an angle from the positive x-axis. The radius tells you how far from the origin the point is, and the angle tells you the direction relative to the positive x-axis.

Convert Polar Coordinates to Cartesian

To plot each point, we need to convert from polar coordinates \((r, \theta)\) to Cartesian coordinates \((x, y)\) using the formulas: \[x = r \cos(\theta)\] and \[y = r \sin(\theta)\].

Calculate Cartesian Coordinates

Let's convert each set of polar coordinates:1. \((3, \frac{1}{3}\pi)\):\[x = 3\cos\left(\frac{1}{3}\pi\right) = 3 \times \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2},\ \ y = 3\sin\left(\frac{1}{3}\pi\right) = 3 \times \frac{1}{2} = \frac{3}{2}\]2. \((1, \frac{1}{2}\pi)\):\[x = 1\cos\left(\frac{1}{2}\pi\right) = 0,\ \ y = 1\sin\left(\frac{1}{2}\pi\right) = 1\]3. \((4, \frac{1}{3}\pi)\):\[x = 4\cos\left(\frac{1}{3}\pi\right) = 2\sqrt{3},\ \ y = 4\sin\left(\frac{1}{3}\pi\right) = 2\]4. \( (0, \pi) \):The radius is 0, hence the Cartesian coordinate is the origin: (0, 0).5. \( (1, 4\pi) \):This is the same as \( (1, 0) \) because angles over \( 2\pi \) wrap around.\[x = 1\cos(0) = 1,\ \ y = 1\sin(0) = 0\]6. \( (3, \frac{11}{7}\pi) \):\[x = 3\cos\left(\frac{11}{7}\pi\right),\ \ y = 3\sin\left(\frac{11}{7}\pi\right)\] (Cyclic properties of trig functions give specific coordinate values; use angle simplification if necessary)7. \( \left( \frac{5}{3}, \frac{1}{2}\pi \right) \):\[x = \frac{5}{3} \cos\left(\frac{1}{2}\pi\right) = 0, \ \ y = \frac{5}{3} \sin\left(\frac{1}{2}\pi\right) = \frac{5}{3}\]8. \( (4, 0) \):\[x = 4 \cos(0) = 4,\ \ y = 4 \sin(0) = 0\]

Plot the Points

Now plot each Cartesian point on graph paper or using a graphing tool. For an accurate plot, mark each coordinate and draw an arrow from the origin to the plotted point to indicate direction.1. \( \left( \frac{3\sqrt{3}}{2}, \frac{3}{2} \right) \)2. \( (0, 1) \)3. \( (2\sqrt{3}, 2) \)4. \( (0, 0) \)5. \( (1, 0) \)6. Custom calculation required for angles like \( \frac{11}{7}\pi \)7. \( (0, \frac{5}{3}) \)8. \( (4, 0) \)

Key concepts

Cartesian coordinates

The Cartesian coordinate system is a way of describing the position of points in space using a pair of numerical coordinates. These coordinates are measured along two perpendicular axes: usually called the x-axis (horizontal) and the y-axis (vertical). Each point in this system is determined by an x-value and a y-value, denoted as \((x, y)\). This system is named after René Descartes, a French mathematician and philosopher who developed analytical geometry.

The strength of Cartesian coordinates is their ability to describe points in a straightforward linear fashion, making it easy to perform calculations and derive formulas. In this approach, points are plotted in a grid layout, creating a rectangular coordinate plane. This system is widely used in various fields, such as physics, engineering, and computer graphics, due to its simplicity and broad applicability.

trigonometric functions

Trigonometric functions, like sine and cosine, are essential for converting between polar and Cartesian coordinates. These functions relate angles in a right triangle to ratios of two of its sides.

• Sine function (\(\sin\)): The sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse.
• Cosine function (\(\cos\)): The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

In the context of polar and Cartesian coordinates, these functions allow for the transformation between the two systems. Given a polar coordinate \((r, \theta)\), the Cartesian x and y can be found using:
\(x = r \cos(\theta)\)
\(y = r \sin(\theta)\)

These transformations are key to solving many problems in mathematics, physics, and engineering that involve rotations and circular paths.

graphing polar coordinates

Graphing polar coordinates requires understanding how to interpret radial and angular data on a plane. Unlike Cartesian coordinates that use a grid system, polar coordinates use a point's radius and angle from a fixed direction (usually the positive x-axis) to determine its position.

To graph polar coordinates, follow these simple steps:

• Start at the origin, the center point of the graph.
• From the origin, measure out the radius of the point along a straight line.
• Then, pivot around the origin by the given angle in a counterclockwise direction to mark the point.

Polar coordinates are especially useful in situations involving rotations and circular motions, such as plotting circles, roses, and spirals. They provide a unique perspective that simplifies calculations and visualizations of rotational dynamics.

polar to Cartesian conversion

Converting polar coordinates to Cartesian coordinates may seem complex at first, but it follows a simple set of rules using trigonometric functions. In polar coordinates, each point is represented by \((r, \theta)\), where \(r\) is the radius or distance from the origin, and \(\theta\) is the angle from the positive x-axis. The conversion formula to Cartesian coordinates involves:

\[x = r \cos(\theta)\]
\[y = r \sin(\theta)\]

These equations ensure you find the precise location of a point in the Cartesian plane.

For example, transforming polar coordinate \((3, \frac{\pi}{3})\) to Cartesian form would result in:
\[x = 3 \times \cos\left(\frac{\pi}{3}\right) = \frac{3\sqrt{3}}{2}\]
\[y = 3 \times \sin\left(\frac{\pi}{3}\right) = \frac{3}{2}\]
This conversion method is essential for interpreting data in multi-dimensional spaces and aids in graphing, modeling, and solving practical math problems efficiently.