Polar Coordinates Parametric Equations Test – Precalculus Math (Printable PDF)

Question 1 Polar to Rectangular (2 points)

Convert (10, 7π6) to rectangular coordinates. Give exact values.

Question 2 Rectangular to Polar (2 points)

Write (−2, 2√3) as (r, θ) with r > 0 and 0 ≤ θ < 2π.

Question 3 Converting Equations (2 points)

Convert each polar equation to rectangular form.

1. r = −4 sin θ (1 point)
2. r = 6 sec θ (1 point)

Question 4 Reading a Rose (2 points)

The figure shows a rose curve of the form r = a cos nθ.

1. State the number of petals and the length of one petal. (1 point)
2. Write an equation of a rose with 7 petals, each 2 units long. (1 point)

Question 5 Cardioid Values (2 points)

For r = 2 + 2 sin θ, find r at θ = 0, π2, π and 3π2. Then state the angle at which the graph touches the pole.

Question 6 Eliminating the Parameter (2 points)

Eliminate the parameter.

1. x = 3t − 2, y = 1 − 2t (1 point)
2. x = 1 + 3 cos t, y = −2 + 3 sin t (1 point)

Question 7 A Thrown Water Balloon (3 points)

A water balloon follows x = 15t and y = 4 + 28t − 16t², in feet, t in seconds.

1. Find its position at t = 1. (1 point)
2. Find its maximum height. (1 point)
3. Find how far it travels horizontally before it lands, to the nearest tenth of a foot. (1 point)

Question 8 Distance on a Radar Screen (3 points)

A radar shows ship A at polar coordinates (8 mi, 20°) and ship B at (5 mi, 80°).

1. Find the angle AOB at the radar O. (1 point)
2. Find the distance AB. Give an exact value. (2 points)

Question 9 Intersecting Circles (2 points)

The figure shows the circles r = 4 and r = 8 sin θ. Find the polar coordinates of both intersection points with r > 0 and 0 ≤ θ ≤ π.