In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point.(4, 90°)

Verified step by step guidance

1. Identify the polar coordinates given: \((r, \theta) = (4, 90^\circ)\).
2. Recall the formulas to convert polar coordinates to rectangular coordinates: \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\).
3. Substitute the given values into the formulas: \(x = 4 \cdot \cos(90^\circ)\) and \(y = 4 \cdot \sin(90^\circ)\).
4. Evaluate \(\cos(90^\circ)\) and \(\sin(90^\circ)\) using known trigonometric values.
5. Calculate the expressions to find the rectangular coordinates \((x, y)\).

Key Concepts

Polar Coordinates

Polar coordinates represent a point in a two-dimensional space using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). In this system, a point is defined by the pair (r, θ), where 'r' is the radial distance and 'θ' is the angle in degrees or radians.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, express a point in a two-dimensional space using two perpendicular axes: the x-axis and the y-axis. A point is represented as (x, y), where 'x' is the horizontal distance from the origin and 'y' is the vertical distance. Converting from polar to rectangular coordinates involves using trigonometric functions.

Conversion from Polar to Rectangular Coordinates

To convert polar coordinates (r, θ) to rectangular coordinates (x, y), the formulas x = r * cos(θ) and y = r * sin(θ) are used. This transformation relies on the relationships defined by the unit circle, where the cosine and sine functions correspond to the x and y coordinates of a point on the circle at a given angle θ.