Full unit practice requiring complete reasoning and representation checks.
Directions: Show exact work before decimals, label domains or intervals, and explain calculator use.
1. {'id': 'u08-l01', 'number': 1, 'title': 'Polar Coordinates and Multiple Representations', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Convert (r,θ)=(4,π/3) to Cartesian coordinates.
2. {'id': 'u08-l01', 'number': 1, 'title': 'Polar Coordinates and Multiple Representations', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Convert (−3,3π/4) to an equivalent point with positive radius.
3. {'id': 'u08-l01', 'number': 1, 'title': 'Polar Coordinates and Multiple Representations', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Write x²+y²=25 in polar form.
4. {'id': 'u08-l01', 'number': 1, 'title': 'Polar Coordinates and Multiple Representations', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Write x=r cos θ and y=r sin θ as the key conversion pair.
5. {'id': 'u08-l01', 'number': 1, 'title': 'Polar Coordinates and Multiple Representations', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Test r=2cosθ for symmetry about the polar axis.
6. {'id': 'u08-l01', 'number': 1, 'title': 'Polar Coordinates and Multiple Representations', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
What curve is r=4sinθ?
7. {'id': 'u08-l02', 'number': 2, 'title': 'Converting Between Polar and Cartesian Forms', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Convert (r,θ)=(4,π/3) to Cartesian coordinates.
8. {'id': 'u08-l02', 'number': 2, 'title': 'Converting Between Polar and Cartesian Forms', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Convert (−3,3π/4) to an equivalent point with positive radius.
9. {'id': 'u08-l02', 'number': 2, 'title': 'Converting Between Polar and Cartesian Forms', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Write x²+y²=25 in polar form.
10. {'id': 'u08-l02', 'number': 2, 'title': 'Converting Between Polar and Cartesian Forms', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Write x=r cos θ and y=r sin θ as the key conversion pair.
11. {'id': 'u08-l02', 'number': 2, 'title': 'Converting Between Polar and Cartesian Forms', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
Test r=2cosθ for symmetry about the polar axis.
12. {'id': 'u08-l02', 'number': 2, 'title': 'Converting Between Polar and Cartesian Forms', 'family': 'polar', 'week': 23, 'unit': 8, 'unit_title': 'Polar Coordinates & Complex Numbers', 'essential': 'How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?', 'project': 'Polar Design Gallery & Complex Rotation Demonstration'}
What curve is r=4sinθ?