Convert (r,θ)=(4,π/3) to Cartesian coordinates.
Answer: (2,2√3)
Reasoning: x=4cosπ/3=2; y=4sinπ/3=2√3.
Reasoning: x=4cosπ/3=2; y=4sinπ/3=2√3.
Teacher Key · Unit 08 · Lesson 01
Polar Coordinates and Multiple Representations
Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.
Khaemenes Academy · Teacher Key
Polar Coordinates and Multiple Representations
Convert (−3,3π/4) to an equivalent point with positive radius.
Answer: (3,7π/4)
Reasoning: Add π to the angle and change the sign of r.
Reasoning: Add π to the angle and change the sign of r.
Write x²+y²=25 in polar form.
Answer: r=5
Reasoning: r²=25 and r≥0 for the circle.
Reasoning: r²=25 and r≥0 for the circle.
Write x=r cos θ and y=r sin θ as the key conversion pair.
Answer: x=r cos θ; y=r sin θ
Reasoning: These project the radius onto the axes.
Reasoning: These project the radius onto the axes.
Test r=2cosθ for symmetry about the polar axis.
Answer: Yes
Reasoning: Replacing θ by −θ leaves the equation unchanged.
Reasoning: Replacing θ by −θ leaves the equation unchanged.
What curve is r=4sinθ?
Answer: A circle
Reasoning: It is a circle of radius 2 centered at (0,2).
Reasoning: It is a circle of radius 2 centered at (0,2).