Polar Coordinates Parametric Equations Test Answer Key with Scoring – Precalculus

Question 1 Polar to Rectangular (2 points)

cos 7π6 = −√32 and sin 7π6 = −12. So x = −5√3 and y = −5.

Answer: (−5√3, −5)

Scoring: 1 point for each coordinate.

Question 2 Rectangular to Polar (2 points)

r = √(4 + 12) = 4. The reference angle is tan⁻¹(√3) = π3 and the point is in Quadrant II, so θ = 2π3.

Answer: (4, 2π3)

Scoring: 1 point for r, 1 point for θ.

Question 3 Converting Equations (2 points)

(a) Multiply by r: r² = −4r sin θ, so x² + y² = −4y and x² + (y + 2)² = 4.

(b) r = 6cos θ means r cos θ = 6, so x = 6.

Answer: (a) x² + (y + 2)² = 4; (b) x = 6

Scoring: 1 point for each part.

Question 4 Reading a Rose (2 points)

(a) The figure has 4 petals, each reaching the 6th ring, so each is 6 units long. (This is r = 6 cos 2θ.)

(b) An odd n gives n petals, so n = 7 and a = 2.

Answer: (a) 4 petals, 6 units each; (b) r = 2 cos 7θ (or 2 sin 7θ)

Scoring: 1 point for part (a), 1 point for part (b).

Question 5 Cardioid Values (2 points)

r(0) = 2, r(π2) = 4, r(π) = 2, r(3π2) = 0.

The graph touches the pole where r = 0, at θ = 3π2.

Answer: 2, 4, 2, 0; the pole at θ = 3π2

Scoring: 1 point for the four values, 1 point for the angle.

Question 6 Eliminating the Parameter (2 points)

(a) t = x + 23, so y = 1 − 2(x + 2)3 = −2x + 13.

(b) cos t = x − 13 and sin t = y + 23, so (x − 1)² + (y + 2)² = 9, a circle with center (1, −2) and radius 3.

Answer: (a) y = −2x + 13; (b) (x − 1)² + (y + 2)² = 9

Scoring: 1 point for each part.

Question 7 A Thrown Water Balloon (3 points)

(a) x = 15 and y = 4 + 28 − 16 = 16. The position is (15, 16).

(b) The vertex is at t = 2832 = 0.875, where y = 4 + 24.5 − 12.25 = 16.25 ft.

(c) Solve 4t² − 7t − 1 = 0: t = 7 + √658 ≈ 1.883 s, so x ≈ 15 · 1.883 ≈ 28.2 ft.

Answer: (15, 16); 16.25 ft; about 28.2 ft

Scoring: 1 point for each part.

Question 8 Distance on a Radar Screen (3 points)

(a) 80° − 20° = 60°.

(b) Law of cosines: d² = 8² + 5² − 2(8)(5) cos 60° = 64 + 25 − 40 = 49, so d = 7.

Answer: 60°; 7 miles

Scoring: 1 point for part (a); 1 point for the law of cosines setup and 1 point for the distance.

Question 9 Intersecting Circles (2 points)

Set 4 = 8 sin θ, so sin θ = 12 and θ = π6 or 5π6.

Answer: (4, π6) and (4, 5π6)

Scoring: 1 point for each point.