X = r cos(t)

Definition

The equation x = r cos(t) represents the relationship between Cartesian and polar coordinates, where 'x' is the horizontal coordinate, 'r' is the distance from the origin to a point in the plane, and 't' (theta) is the angle measured from the positive x-axis. This formula is essential for converting polar coordinates to Cartesian coordinates, allowing for the analysis of curves described in polar form.

5 Must Know Facts For Your Next Test

1. In polar coordinates, 'r' can be thought of as the hypotenuse of a right triangle formed by the x and y components.
2. The cosine function in x = r cos(t) gives the horizontal component of the vector formed by r at an angle t.
3. This equation shows that as 't' changes, 'x' varies depending on both the angle and the radius 'r'.
4. If t = 0, then x = r, placing the point directly on the positive x-axis at a distance r from the origin.
5. Converting polar coordinates using x = r cos(t) is useful for graphing and analyzing functions that are easier to express in polar form.

Review Questions

• How does the equation x = r cos(t) illustrate the relationship between polar and Cartesian coordinates?
  • The equation x = r cos(t) illustrates how polar coordinates can be transformed into Cartesian coordinates. In this case, 'r' represents the radial distance from the origin, and 't' indicates the angle from the positive x-axis. By applying this formula, we can determine the x-coordinate of a point based on its radial distance and angle, effectively bridging the two coordinate systems.
• Discuss how changes in angle 't' affect the value of 'x' in the equation x = r cos(t).
  • In the equation x = r cos(t), as 't' varies, it impacts 'x' based on the cosine function's behavior. Since cosine ranges from 1 to -1, changing 't' alters how far along the x-axis we project our point. For example, when t is 0, x equals r; when t is π/2, x equals 0; and as t approaches π, x becomes negative. This shows how angles influence positioning in a plane.
• Evaluate how understanding x = r cos(t) can enhance our ability to analyze complex curves in polar coordinates.
  • Understanding x = r cos(t) allows us to convert complex curves expressed in polar coordinates into Cartesian form for easier analysis and graphing. By expressing polar curves through this relationship, we can examine their properties more thoroughly, such as intercepts and symmetry. This insight not only aids in visualization but also enhances our capability to apply calculus concepts like derivatives and integrals to polar functions.

Related terms

Polar Coordinates: A coordinate system where each point is determined by a distance from a reference point (the pole) and an angle from a reference direction.

Cartesian Coordinates: A coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, typically denoted as (x, y).

radians: A unit of angular measurement in which angles are defined based on the radius of a circle, with one complete revolution equaling 2π radians.