Polar Coordinates and Graphs

Develop radius and angle representations, converting and reading polar curves.

Lesson 1 of 32 parts Practice Flashcards

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Radius and Angle Representations

Objective: Explain radius and angle representations, apply the rule, and justify relevant restrictions.

A polar coordinate pair (r,θ) locates a point by signed radius and angle. Cartesian conversion is x=r cos θ and y=r sin θ. A positive radius is √(x²+y²); choose the angle using both signs of x and y. The representations (r,θ+2πk) and (−r,θ+π+2πk) describe the same point. At the origin, the angle is not unique.

Formulas

$x=r cos θ; y=r sin θ; r²=x²+y²$

Worked examples

Try itA point has Cartesian coordinates (21,28). Find its positive polar radius.Show answer

Step 1: Use r=√(x²+y²). Step 2: Compute √(441+784). Step 3: The positive radius is 35; θ belongs to quadrant I.

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Radius and Angle Representations

Objective: Explain radius and angle representations, apply the rule, and justify relevant restrictions.

Formulas

$x=r cos θ; y=r sin θ; r²=x²+y²$

Worked examples

Try itA point has Cartesian coordinates (21,28). Find its positive polar radius.Show answer

Step 1: Use r=√(x²+y²). Step 2: Compute √(441+784). Step 3: The positive radius is 35; θ belongs to quadrant I.

Converting and Reading Polar Curves

Objective: Explain converting and reading polar curves, apply the rule, and justify relevant restrictions.

For a polar equation r=f(θ), each parameter angle supplies a radius and hence a point. Convert with x=r cos θ and y=r sin θ; negative r plots across the origin. A full curve can require an interval other than one turn, depending on the rule. For r=2a cos θ, multiplying by r gives x²+y²=2ax, a circle, while avoiding division that could lose the origin.

Formulas

$r=2a cos θ ⇒ x²+y²=2ax$

Worked examples

Try itA polar point has r=14 and θ=π/3. Find x.Show answer

Step 1: Use x=r cos θ. Step 2: cos(π/3)=1/2, so x=(14)(1/2). Step 3: The x-coordinate is 7.

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