Question

Write in rectangular form. $$6 /-125^{\circ}$$

Short answer

The rectangular form of \( 6 /-125^{\text{o}} \) is approximately \( (-3.4416, -4.9146) \).

Step-by-step solution

Understanding Polar to Rectangular Conversion

To convert the polar form to rectangular form, use the conversion formulas: \( x = r \times \text{cos}(\theta) \) and \( y = r \times \text{sin}(\theta) \), where \( r \) is the magnitude and \( \theta \) is the angle in degrees.

Apply the Conversion Formulas

For the given polar coordinate \( 6 /-125^{\text{o}} \), calculate the rectangular coordinates \( x \) and \( y \) by using the formulas: \( x = 6 \times \text{cos}(-125^{\text{o}}) \) and \( y = 6 \times \text{sin}(-125^{\text{o}}) \).

Calculate the Cosine and Sine Values

Using a calculator, find the cosine and sine of \( -125^{\text{o}} \). Remember to use the negative angle if indicated. \( \text{cos}(-125^{\text{o}}) \approx -0.5736 \) \( \text{sin}(-125^{\text{o}}) \approx -0.8191 \)

Compute Rectangular Coordinates

Multiply the magnitude \( 6 \) by the values found in the previous step: \( x = 6 \times (-0.5736) = -3.4416 \) \( y = 6 \times (-0.8191) = -4.9146 \)

Write the Rectangular Form

Combine the \( x \) and \( y \) values to form the rectangular coordinate: \( (x, y) = (-3.4416, -4.9146) \)

Key concepts

Rectangular Coordinates

Rectangular coordinates are a way of representing points in a plane using two numbers, often referred to as the x (horizontal) and y (vertical) axes. This system, also known as Cartesian coordinates, is fundamental to fields such as geometry, algebra, and calculus. Imagine a rectangular grid where each point has a unique position determined by its distance from the origin (0,0) along these axes.

For instance, in a 2D coordinate system, a point (x, y) represents the horizontal movement from the origin by x units and a vertical movement by y units. To locate a point, one would move x units along the horizontal axis and then y units up or down, creating a rectangular shape hence the name. This coordinate system is widely used for its simplicity and ability to easily relate to physical space.

Polar Coordinates

Contrary to rectangular coordinates, polar coordinates define positions based on a distance from a central point, known as the pole, and an angle relative to a directional line, usually the positive x-axis. A point in polar coordinates is represented as (r, θ), where 'r' denotes the radial distance from the pole, and 'θ' represents the angle in degrees or radians.

These coordinates are particularly useful in situations involving circular or spiral motions, and patterns where the relationship between points is better described through angles and distances. Trigonometry often uses polar coordinates when dealing with periodic functions, while fields such as physics and engineering use them to solve problems with rotational symmetry.

Trigonometry

Trigonometry is a branch of mathematics concerning the study of triangles, specifically right-angled triangles, and the relationships between their angles and sides. At the heart of trigonometry are the sine, cosine, and tangent functions which relate the angles of a triangle to the ratios of its sides.

These functions are critical for coordinate conversion because they can be used to describe the relationship between the angle of a vector (direction) and its horizontal and vertical components. For example, the cosine of an angle in a right triangle represents the ratio of the adjacent side to the hypotenuse, while the sine represents the ratio of the opposite side to the hypotenuse. Understanding these relationships is crucial for converting between polar and rectangular coordinates.

Coordinate Conversion

Coordinate conversion is the process of translating points from one coordinate system to another. In our context, that means converting polar coordinates (r, θ) into rectangular coordinates (x, y) and vice versa. The process uses trigonometric functions to enable this transition.

The formulas for converting from polar to rectangular coordinates are: \[ x = r \times \cos(\theta) \] and \[ y = r \times \sin(\theta) \]. Conversely, to convert from rectangular to polar coordinates, the formulas are: \[ r = \sqrt{x^2 + y^2} \] and \[ \theta = \text{atan2}(y, x) \], where atan2 is a variant of the arctangent function that takes into account the signs of both x and y to determine the correct quadrant for θ.