Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Work from direct conversions to projectile, radar and Ferris wheel models. Give exact values with π when possible.
Standards: N-CN.B.4, F-IF.C.7 · Level: Grade 12 Precalculus
Problem 1 Polar to Rectangular ★★★
Convert the polar point (6, 60°) to rectangular coordinates.
Problem 2 A Negative Radius ★★★
Convert the polar point (−3, π4) to rectangular coordinates.
Problem 3 Rectangular to Polar ★★★
Write (−3√3, 3) in polar form with r > 0 and 0 ≤ θ < 2π.
Problem 4 A Point in Quadrant III ★★★
Write (−6, −6) in polar form with r > 0 and 0 ≤ θ < 2π.
Problem 5 Reading a Polar Grid ★★★
Point A is shown on the polar grid. (a) Write A as (r, θ) with r > 0. (b) Give two other polar names for A. (c) Find its rectangular coordinates.
Problem 6 A Circle from a Sine ★★★
Convert r = 8 sin θ to rectangular form and describe the graph.
Problem 7 A Line in Polar Form ★★★
Write the line 2x − 5y = 10 in polar form, solving for r.
Problem 8 A Circle Through the Pole ★★★
Convert x2 + y2 + 6x = 0 to polar form, solving for r (ignore the pole).
Problem 9 Analyzing a Cardioid ★★★
Consider r = 3 + 3 cos θ. (a) Name the type of graph. (b) Find r at θ = 0, π2, π, 3π2. (c) State the farthest distance from the pole and where the graph touches the pole.
Problem 10 An Eight-Petal Rose ★★★
Describe the graph of r = 5 sin 4θ: the number of petals, the length of each petal, and the first angle at which a petal tip occurs for θ ≥ 0.
Problem 11 A Limaçon with a Loop ★★★
Show that r = 2 + 4 cos θ has an inner loop. Find the angles where the graph passes through the pole and the largest value of r.
Problem 12 A Three-Petal Rose ★★★
Find the tip of each petal of r = 4 cos 3θ for 0 ≤ θ < π, and state how many petals the graph has.
Problem 13 Eliminating a Parameter ★★★
A particle moves with x = 2t + 1 and y = t − 3 for 0 ≤ t ≤ 5. Eliminate the parameter, then give the starting and ending points.
Problem 14 An Elliptical Path ★★★
A drone circles a pond along x = 2 + 5 cos t, y = −1 + 3 sin t, for 0 ≤ t < 2π (distances in yards). Find the rectangular equation of the path, its center, and its semi-axes.
Problem 15 Is It a Parabola? ★★★
Convert r = 41 + cos θ to rectangular form and identify the graph.
Problem 16 The Kicked Ball ★★★
A ball is kicked with x = 18t and y = 5 + 24t − 16t2, in feet, with t in seconds. (a) Find the ball’s position at t = 0.5. (b) Find its maximum height. (c) Find how far it travels horizontally before hitting the ground, to the nearest tenth of a foot.
Problem 17 Two Boats on Radar ★★★
A radar screen shows boat A at polar coordinates (9 mi, 30°) and boat B at (7 mi, 150°). (a) Find the angle at the radar between the two boats. (b) Find the distance between them to the nearest tenth of a mile.
Problem 18 Microphone Pickup Pattern ★★★
A studio microphone picks up sound in the pattern r = 4 + 4 sin θ (in feet of effective range), where θ = π2 points straight at the performer. (a) Name the shape. (b) Find the range straight ahead and straight behind (θ = 3π2). (c) Find the range at θ = π6.
Problem 19 Where Two Curves Meet ★★★
Find the points where r = 3 and r = 6 cos θ intersect. Give them in polar form for −π < θ ≤ π, then in rectangular form.
Problem 20 Ferris Wheel Motion ★★★
A rider on a Ferris wheel moves with x = 20 cos(t5) and y = 24 + 20 sin(t5), in feet, t in seconds. (a) Eliminate the parameter. (b) Find the rider’s position at t = 5 to the nearest tenth. (c) Find the time of one full turn.
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