Question

Convert from rectangular coordinates (x,y) to polar coordinates (r,θ) using the conversion formulas.

r=√x²+y²
θ=tan⁻¹(y/x)

Answer 1

To convert from rectangular to polar coordinates, we use the formulas to calculate the radial distance r and the angle θ. This process involves finding the square root of the sum of the squares of x and y for r, and the arctangent of y/x for θ.

Step-by-step explanation:

The student's question pertains to the conversion from rectangular coordinates to polar coordinates, which involves two steps. First, the radial distance r is calculated using the formula r = √x²+y². This represents the distance from the origin to the point in the plane. The second step involves finding the angle θ (theta) using the arctangent function: θ = tan⁻¹(y/x). It's important to note that the arctangent function can have two results differing by π radians, so the correct angle must be chosen based on the quadrant in which the point lies.

Using these formulas, the polar coordinates can be determined by substituting the given x and y values from the rectangular coordinates. For an example, if we have a point with rectangular coordinates (3, 4), we compute the polar coordinates as follows:

1. Calculate r: r = √ (3² + 4²) = 5.
2. Calculate θ: θ = tan⁻¹ (4/3), which is approximately 0.927 radians.

The polar coordinates would be (5, 0.927 radians).