Question

Write the equations that are used to express a point with polar coordinates \((r, \theta)\) in Cartesian coordinates.

Short answer

Question: Write the equations used to express a point with polar coordinates (r, θ) in Cartesian coordinates (x, y). Answer: The equations to express a point with polar coordinates (r, θ) in Cartesian coordinates (x, y) are: \(x = r \cos(\theta)\) \(y = r \sin(\theta)\)

Step-by-step solution

Understand the relationship between polar and Cartesian coordinates

In polar coordinates, a point is represented by two values: the radius \(r\), which is the distance from the origin to the point, and the angle \(\theta\), which is the angle between the positive x-axis and the line connecting the origin to the point. In Cartesian coordinates, a point is represented by two values: the x-coordinate and the y-coordinate. Our goal is to find equations that convert polar coordinates (r, θ) to Cartesian coordinates (x, y).

Express the x-coordinate in terms of polar coordinates

Using the angle θ, and the radius r, we can create a right triangle where the base (x-coordinate) and the height (y-coordinate) are the legs, and r is the hypotenuse. The cosine function relates the angle, base, and hypotenuse in a right triangle. Therefore, we can write the x-coordinate as: \[x = r \cos(\theta)\]

Express the y-coordinate in terms of polar coordinates

Similarly, we can use the sine function to relate the angle, height, and hypotenuse in a right triangle. Therefore, we can write the y-coordinate as: \[y = r \sin(\theta)\] Now, we have the two equations to express a point with polar coordinates (r, θ) in Cartesian coordinates (x, y): \[x = r \cos(\theta)\] \[y = r \sin(\theta)\]

Key concepts

Polar Coordinates

Polar coordinates are a way to describe the position of a point in a plane using a radius and an angle. Unlike Cartesian coordinates, which use horizontal and vertical distances (x and y) from the origin, polar coordinates use the distance from the origin and the angle from the positive x-axis.
This angle is typically measured in radians. The radius, denoted as \(r\), tells us how far the point is from the origin, and \(\theta\) describes the direction of this distance.

• \(r\) is always non-negative.
• \(\theta\) represents the angle made with the positive x-axis.

Polar coordinates are especially useful in scenarios involving circular or rotational patterns because they naturally align with these shapes.

Cartesian Coordinates

Cartesian coordinates are the most common way to represent points in a plane. Named after the mathematician René Descartes, these coordinates define a point through two perpendicular axes: x and y.
The x-coordinate tells how far left or right a point is from the origin, while the y-coordinate tells how far up or down it is.

• x-coordinate: Horizontal distance from the origin.
• y-coordinate: Vertical distance from the origin.

Together, these values give a precise location of any point in the two-dimensional plane and are very useful in algebraic and geometric applications.

Trigonometric Functions

Trigonometric functions are essential for understanding the relationships in a right triangle involving angles and lengths of sides. In polar to Cartesian conversion, two key functions are used: cosine and sine.
These functions help translate the angle and radius from polar coordinates into the horizontal and vertical distances in the Cartesian system.

• Cosine function \(\cos(\theta)\): Relates the angle to the adjacent side over hypotenuse.
• Sine function \(\sin(\theta)\): Relates the angle to the opposite side over hypotenuse.

For a point \((x, y)\):
- \(x = r \cos(\theta)\)
- \(y = r \sin(\theta)\)
These functions bridge the gap between the polar and Cartesian systems.

Coordinate Transformation

Coordinate transformation involves changing the representation of a point from one coordinate system to another. It is a powerful tool in mathematics and physics, allowing us to switch perspectives and simplify problems.
When converting from polar to Cartesian coordinates, we use trigonometric functions to map a point described by \((r, \theta)\) to \((x, y)\).

• The x-coordinate is found using \(x = r \cos(\theta)\).
• The y-coordinate is found using \(y = r \sin(\theta)\).

These equations allow us to translate problems from circular systems to linear systems or vice-versa, making them easier to solve depending on the context.