Convert Complex Numbers from Polar to Cartesian (r, θ → x, y)

This example visually demonstrates how a complex number’s polar form (with modulus r and argument θ) corresponds to its Cartesian coordinates (x and y) on the coordinate plane. It helps learners see that the same complex number can be represented geometrically in different forms: z=r(cosθ+i sinθ), and equivalently x=r cosθ, y=r sinθ. The construction centers on the relationship between the radius segment from the origin, the marked angle θ, and the right-triangle/trigonometric interpretation of the coordinate projections. In GeoGebra, the build uses Translate (to position points and labels), FractionText (for nicely formatted formula text), Segment (to draw the radius/connection lines such as OP), and Angle (to display and manage the argument θ). Key objects include r and theta, points O and P, the complex label z_{1}, and text/angle decoration objects (e.g., alphaDecor, txtAngle) that link the visual geometry to the algebraic expressions. Suitable for Grade 11–level complex numbers, coordinate geometry, and trigonometry instruction, as well as for classroom demonstrations and guided self-study.