Question

Convert the point (3, 3) from Cartesian coordinates to polar coordinates, where r = 20 and θ < 2π.

Answer 1

To convert the Cartesian coordinates (3, 3) to polar coordinates where r = 20 and θ < 2π, we find θ using arctan(y/x), which results in θ = π/4. So, the polar coordinates are (20, π/4).

Step-by-step explanation:

To convert the Cartesian coordinates (3, 3) into polar coordinates with a fixed radius r = 20, we need to determine the angle θ. We use the formulas for converting to polar coordinates:

• x = r × cos(θ)
• y = r × sin(θ)

We have x and y, and we are given r, so we solve for θ using these relationships. For (3, 3), the angle θ in radians can be found by the inverse trigonometric functions, particularly arctan(y/x), as long as θ is less than 2π and r is kept as 20.

θ = arctan(3/3) = arctan(1)

Since arctan(1) corresponds to π/4 (or 45°), and we need θ to be less than 2π, θ = π/4 satisfies this condition. Therefore, the polar coordinates are (20, π/4).