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Convert between polar and rectangular coordinates, graph polar equations, and understand polar curves.
Learn step-by-step with practice exercises built right in.
In the polar coordinate system, each point is determined by:
A point is written as in polar form.
Convert the point from polar to rectangular coordinates.
Avoid these 4 frequent errors
See how this math is used in the real world
A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of cm/s. How fast is the area of the circle increasing when the radius is cm?
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Given in polar:
Given in rectangular: (adjust for quadrant)
Or use:
Important: When finding :
General form: or
General form: or
Test for symmetry to help sketch graphs:
Symmetry about the polar axis (x-axis):
Symmetry about (y-axis):
Symmetry about the pole (origin):
Solution:
Given polar coordinates:
Use conversion formulas:
Find :
Find :
Answer:
Verification:
Convert the rectangular equation to polar form.
Solution:
Given:
Use substitutions:
Identify and sketch the polar curve . Describe its key features.
Solution:
Given:
Step 1: Identify the curve type
This is a limaçon of the form where and .
Since , this is a (heart-shaped).
Step 2: Check for symmetry
Test symmetry about the polar axis (x-axis): Replace with :
Equation unchanged, so symmetric about the polar axis ✓
Step 3: Create table of values
Step 4: Key features
Sketch description: The curve starts at , curves upward and left, passes through , continues to the origin at forming a cusp, then curves downward through , and returns to . The overall shape resembles a heart lying on its side.
Convert the polar coordinates (4, π/3) to rectangular coordinates.
Step 1: Use conversion formulas: x = r cos(θ) y = r sin(θ)
Step 2: Identify r and θ: r = 4, θ = π/3
Step 3: Calculate x: x = 4 cos(π/3) = 4 · (1/2) = 2
Step 4: Calculate y: y = 4 sin(π/3) = 4 · (√3/2) = 2√3
Step 5: Write rectangular coordinates: (x, y) = (2, 2√3)
Answer: (2, 2√3)
Convert the rectangular equation x² + y² = 9 to polar form.
Step 1: Recall conversion relationships: x = r cos(θ) y = r sin(θ) x² + y² = r²
Step 2: Substitute into equation: x² + y² = 9 r² = 9
Step 3: Solve for r: r = ±3
Step 4: Simplify (standard form): In polar coordinates, we typically use r = 3 (r = -3 represents the same circle)
Step 5: Interpretation: This is a circle with radius 3 centered at the origin
Answer: r = 3
Substitute:
Simplify:
This gives or
Since is included in when or , we can write:
Answer:
Interpretation: This is a circle with diameter 4 on the line (the y-axis).
Verification in rectangular: