X = r cos(θ)

Definition

The equation x = r cos(θ) is a fundamental relationship in polar coordinates that describes the x-coordinate of a point in a polar coordinate system. It expresses the horizontal position of a point in terms of the radial distance (r) and the angle (θ) from the positive x-axis.

5 Must Know Facts For Your Next Test

1. The equation x = r cos(θ) allows you to convert between polar and Cartesian coordinates, as it provides the x-coordinate of a point given its polar coordinates (r, θ).
2. The cosine function in the equation represents the horizontal component of the radial distance, which is the x-coordinate of the point in the Cartesian plane.
3. The angle θ is measured counterclockwise from the positive x-axis, and its cosine value determines the horizontal position of the point.
4. Knowing the equation x = r cos(θ) is crucial for understanding the relationship between polar and Cartesian coordinates and for performing various calculations and transformations.
5. The equation x = r cos(θ) is one of the fundamental equations that defines the conversion between polar and Cartesian coordinate systems.

Review Questions

• Explain how the equation x = r cos(θ) relates the polar coordinates (r, θ) to the x-coordinate of a point in the Cartesian plane.
  • The equation x = r cos(θ) establishes a direct relationship between the polar coordinates (r, θ) and the x-coordinate of a point in the Cartesian plane. The radial distance (r) represents the distance of the point from the origin, while the angle (θ) measures the counterclockwise rotation from the positive x-axis. The cosine function in the equation determines the horizontal component of the radial distance, which corresponds to the x-coordinate of the point. This equation is crucial for converting between polar and Cartesian coordinate systems and understanding the spatial relationships in a polar coordinate system.
• Describe how the angle θ affects the value of x = r cos(θ).
  • The angle θ in the equation x = r cos(θ) plays a significant role in determining the value of the x-coordinate. As the angle θ increases from 0° to 360° in a counterclockwise direction, the cosine function oscillates between 1 and -1, causing the x-coordinate to vary accordingly. When θ is 0°, the cosine function is 1, and the x-coordinate is equal to the radial distance (r). As θ increases, the cosine function decreases, resulting in a smaller x-coordinate. At θ = 90°, the cosine function is 0, and the x-coordinate becomes 0. The cyclical nature of the cosine function allows the x-coordinate to alternate between positive and negative values as the angle θ changes, reflecting the position of the point in the Cartesian plane.
• Analyze how the equation x = r cos(θ) can be used to determine the position of a point in a polar coordinate system relative to the Cartesian coordinate system.
  • The equation x = r cos(θ) provides a direct link between the polar and Cartesian coordinate systems, allowing you to determine the position of a point in the Cartesian plane based on its polar coordinates. By substituting the given values of r (radial distance) and θ (angle) into the equation, you can calculate the x-coordinate of the point. This information, combined with the equation y = r sin(θ) for the y-coordinate, enables you to precisely locate the position of the point in the Cartesian plane. Understanding the relationship between these coordinate systems and the use of the x = r cos(θ) equation is essential for performing various geometric and analytical tasks, such as transforming between coordinate systems, calculating distances and angles, and visualizing the spatial properties of objects in a polar coordinate system.

Related terms

Polar Coordinates:

A coordinate system that uses the distance from a fixed point (the origin) and the angle from a fixed direction (the positive x-axis) to locate a point on a plane.

Radial Distance (r): The distance of a point from the origin in a polar coordinate system.

Angle (θ): The angle between the positive x-axis and the line segment connecting the origin to a point in a polar coordinate system.