25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.


(4, 5π)

Verified step by step guidance

1. Recall the formulas to convert polar coordinates \((r, \theta)\) to Cartesian coordinates \((x, y)\): \(x = r \cos(\theta)\) \(y = r \sin(\theta)\)
2. Identify the given polar coordinates: \(r = 4\) and \(\theta = 5\pi\).
3. Substitute the values into the conversion formulas: \(x = 4 \cos(5\pi)\) \(y = 4 \sin(5\pi)\)
4. Use the periodic properties of trigonometric functions to simplify \(\cos(5\pi)\) and \(\sin(5\pi)\). Remember that \(\cos(\theta)\) and \(\sin(\theta)\) have periods of \$2\pi$.
5. Calculate the simplified values of \(x\) and \(y\) to express the Cartesian coordinates corresponding to the given polar coordinates.

Key Concepts

Polar Coordinates

Polar coordinates represent a point in the plane using a radius and an angle, denoted as (r, θ), where r is the distance from the origin and θ is the angle measured from the positive x-axis. Understanding this system is essential for converting to Cartesian coordinates.

Conversion Formulas from Polar to Cartesian

To convert polar coordinates (r, θ) to Cartesian coordinates (x, y), use the formulas x = r cos(θ) and y = r sin(θ). These formulas translate the radius and angle into horizontal and vertical distances on the Cartesian plane.

Angle Measurement and Trigonometric Values

Angles in polar coordinates are often given in radians. Knowing how to evaluate trigonometric functions like sine and cosine at specific angles, such as multiples of π, is crucial for accurate conversion to Cartesian coordinates.