Problem 92 - Algebra and Trigonometry

Question

Explain how to convert a point from rectangular to polar coordinates. Provide an example with your explanation.

Step-by-Step Solution

Answer

To convert a point from rectangular to polar coordinates: find r using \( r = \sqrt{x^2 + y^2} \), and find θ using \( θ = tan^{-1}(y/x) \). For example, the rectangular coordinates (3, 4) convert to polar coordinates as approximately (5, 53.13°).

Step 1: Understand the formulas

Two main formulas relate rectangular and polar coordinates. They are \( r = \sqrt{x^2 + y^2} \) and \( θ = tan^{-1}(y/x) \). r represents the distance of the point from the origin and θ is the angle made with the positive x-axis.

Step 2: Example - Convert the point (3, 4) from rectangular to polar coordinates

Firstly, to find r, substitute x as 3 and y as 4 in the formula for r. That comes out as \( r = \sqrt{3^2 + 4^2} = 5 \). Secondly, to find θ, substitute y as 4 and x as 3 in the formula for θ. That results in \( θ = tan^{-1}(4/3) \), the value of which depends on the unit (degrees or radians) one needs the answer in. Assuming θ is required in degrees, \( θ = tan^{-1}(4/3) \approx 53.13° \).

Step 3: Present the result

After converting, the rectangular coordinates (3, 4) have the polar coordinates approximately as (5, 53.13°).