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. Recall the relationship between polar coordinates \((r, \theta)\) and rectangular coordinates \((x, y)\), which is given by the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).
2. Identify the given polar coordinates: \(r = 4\) and \(\theta = 90^\circ\).
3. Convert the angle \(\theta\) from degrees to radians if necessary, but since trigonometric functions can be evaluated directly in degrees, you can use \(90^\circ\) as is.
4. Calculate the rectangular coordinates using the formulas: \(x = 4 \cos(90^\circ)\) and \(y = 4 \sin(90^\circ)\).
5. Evaluate the cosine and sine values for \(90^\circ\) and substitute them back into the expressions for \(x\) and \(y\) to find the rectangular coordinates.

Key Concepts

Polar Coordinates

Polar coordinates represent a point in the plane using a distance from the origin (radius) and an angle measured from the positive x-axis. The format is (r, θ), where r is the radius and θ is the angle in degrees or radians.

Conversion from Polar to Rectangular Coordinates

To convert polar coordinates (r, θ) to rectangular coordinates (x, y), use the formulas x = r * cos(θ) and y = r * sin(θ). This translates the point from a radius and angle to Cartesian x and y values.

Trigonometric Functions and Angle Measurement

Understanding sine and cosine functions is essential, as they relate angles to ratios of sides in right triangles. Also, knowing how to work with angles in degrees or radians ensures correct evaluation of these functions during conversion.