Question

Convert \((-8,-8)\) into polar coordinates and \(\left(4, \frac{2 \pi}{3}\right)\) into rectangular coordinates.

Short answer

(-8,-8) is (8√2, 5π/4) in polar; (4, 2π/3) is (-2, 2√3) in rectangular.

Step-by-step solution

Understanding Polar and Rectangular Coordinates

Rectangular coordinates are described by two values, \(x\) and \(y\), representing a position on a grid.Polar coordinates have two values, r and \(\theta\), with \(r\) as the distance from the origin and \(\theta\) as the angle from the positive \(x\)-axis.To solve, we will convert from polar to rectangular and vice-versa.

Convert Rectangular to Polar

Given the point \((-8,-8)\) in rectangular coordinates.Calculate the magnitude, \(r\), using the formula \(r = \sqrt{x^2 + y^2}\).First, compute: \((-8)^2 + (-8)^2 = 64 + 64 = 128\).Next, take the square root: \(r = \sqrt{128} = 8\sqrt{2}\).Next, find the angle, \(\theta\), using \(\theta = \arctan \left(\frac{y}{x}\right)\).Both \(x\) and \(y\) are negative, so \(\theta \approx \arctan(1)\).Thus, \(\theta \approx \frac{5\pi}{4} \) (adding \(\pi\) since the point is in the third quadrant).

Verify Polar Coordinate Result

The point \((-8,-8)\) in polar coordinates transforms to \((8\sqrt{2}, \frac{5\pi}{4})\).Recalculate as necessary to ensure accuracy.

Convert Polar to Rectangular

Start with polar coordinates \((4, \frac{2\pi}{3})\).Determine \(x\) with \(x = r \cos \theta\).Calculate \(\cos \left(\frac{2\pi}{3}\right) = -\frac{1}{2}\).Thus, \(x = 4 \cdot -\frac{1}{2} = -2\).Determine \(y\) with \(y = r \sin \theta\).Calculate \(\sin \left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}\).Thus, \(y = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}\).

Verify Rectangular Coordinate Result

The polar coordinates \((4, \frac{2\pi}{3})\) change to rectangular coordinates \((-2, 2\sqrt{3})\).Consider recalculations if conflicts arise.

Key concepts

Polar Coordinates

Polar coordinates provide a different way to locate points on a plane. Instead of using two perpendicular axes like the rectangular coordinate system, we use:

• Radius: denoted as \(r\), it measures how far a point is from the origin (or center of the graph).
• Angle: denoted as \(\theta\), it indicates the direction in relation to the positive x-axis, measured in radians or degrees.

This system is particularly useful in contexts where the relationship between points is more naturally expressed in terms of angles and distances, like circular and rotational dynamics. Converting between polar and rectangular involves using trigonometric functions to find angles and position effectively.

Rectangular Coordinates

In rectangular coordinates, points are identified based on their horizontal and vertical positions relative to a fixed point called the origin. This system uses:

• x-coordinate: shows the distance left or right from the origin.
• y-coordinate: shows the distance up or down from the origin.

Imagine a grid or a chessboard; each square is demarcated by an \(x\) and a \(y\) position. Rectangular coordinates are intuitive for many problems, especially those involving straight lines. They simplify finding distances and slopes between points using algebraic methods.

Trigonometric Functions

Trigonometric functions are crucial when converting between coordinate systems. They relate angles to side lengths in right triangles, and commonly used ones include:

• Sine (\(\sin\)): \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).
• Cosine (\(\cos\)): \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\).
• Tangent (\(\tan\)): \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).

These functions are essential tools in converting a point from polar to rectangular, and vice versa. For example, calculating \(x\) involves \(\cos\theta\) and calculating \(y\) involves \(\sin\theta\) when moving from polar to rectangular.

Distance Formula

The distance formula is a straightforward way to calculate the distance between two points in the rectangular coordinate system. It is derived from the Pythagorean theorem and is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula helps determine the direct length of a line segment connecting points \((x_1, y_1)\) and \((x_2, y_2)\) in the plane, allowing us to assess spatial relationships. In polar coordinates, the distance to the origin is directly \(r\), simplifying some calculations as you only need to find \(r\) using \(\sqrt{x^2 + y^2}\) when converting from rectangular coordinates.