Polar–Rectangular Conversion
Switch between coordinate systems.
Key idea
Switch between coordinate systems.
In depth
x = r cos θ, y = r sin θ, and r = √(x² + y²), tan θ = y/x. Pick θ in the correct quadrant after finding it with arctan.
Worked example
(−1, 1): r = √2, θ = 135° (not −45°). In polar form (√2, 3π/4).
Formula
Real-world uses
- Switching between map grids and polar radar displays
- Converting code coordinates
Common mistake
Taking arctan(y/x) blindly, which loses the quadrant.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Polar–Rectangular Conversion?
Switch between coordinate systems. It is a topic at the advanced level in Trigonometry.
What is the formula for Polar–Rectangular Conversion?
The key formula is x = r cosθ, y = r sinθ, r² = x²+y².
Where is Polar–Rectangular Conversion used in real life?
Real-world examples: Switching between map grids and polar radar displays; Converting code coordinates.
What should I learn before Polar–Rectangular Conversion?
Make sure you are comfortable with Polar Coordinates, the previous topic in Trigonometry.
What is a common mistake with Polar–Rectangular Conversion?
Taking arctan(y/x) blindly, which loses the quadrant.
How can I practise Polar–Rectangular Conversion?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Polar–Rectangular Conversion?
Move on to Polar Graphs: Circles and Roses next.
Related advanced topics in Trigonometry
- Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
- Phase Shift – Horizontal translation of a wave.
- Vertical Shift and Midline – Moves the wave up or down.
- General Sinusoid – Combine amplitude, period, phase and shift.
- Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
- Pythagorean Identity – The key identity from the unit circle.