CONVERTING BETWEEN POLAR AND RECTANGULAR EQUATIONS WORKSHEET

Problems 1-10 : Convert each polar equation to rectangular form.

Problem 1 :

r = 2

Problem 2 :

tan θ = 2

Problem 3 :

Problem 4 :

r sin θ = 3

Problem 5 :

r = 2cos θ

Problem 6 :

r = 4cos θ - 4sin θ

Problem 7 :

r = 2cos θ + 2sin θ

Problem 8 :

Problem 9 :

Problem 10 :

r² = 5 sec (2θ)

Problems 11-16 : Convert each rectangular equation to polar form.

Problem 11 :

x = y²

Problem 12 :

y = x²

Problem 13 :

2x - 5y = 4

Problem 14 :

x² + y² = 9

Problem 15 :

x² + y² - 7y = 0

Problem 16 :

(x - 1)² + (y + 1)² = 2

Answers

1. Answer :

r = 2

Squaring both sides,

r² = 4

Substitute x² + y² for r².

x² + y² = 4

2. Answer :

tan θ = 2

Multiply both sides by x.

y = 2x

3. Answer :

Take tan on both sides.

4. Answer :

r sin θ = 3

Substitute y for r sin θ.

y = 3

5. Answer :

r = 2cos θ

Multiply both sides by r.

r² = 2rcos θ

Substitute x² + y² for r² and x for r cos θ.

x² + y² = 2x

x² + y² = 2x

x² - 2x + y² = 0

x² - 2(x)(1) + y² = 0

x² - 2(x)(1) + 1² - 1² + y² = 0

(x - 1)² - 1² + y² = 0

(x - 1)² - 1 + y² = 0

(x - 1)² + (y - 0)² = 1

6. Answer :

r = 4cos θ - 4sin θ

Multiply both sides by r.

r² = r(4cos θ - 4sin θ)

r² = 4rcos θ - 4rsin θ

Substitute x² + y² for r², x for r cos θ and y for r sin θ.

x² + y² = 4x - 4y

x² - 4x + y² + 4y = 0

x² - 2(x)(2) + y² + 2(y)(2) = 0

x² - 2(x)(2) + 2² - 2² + y² + 2(y)(2) + 2² - 2² = 0

(x - 2)² - 2² + (y + 2)² - 2² = 0

(x - 2)² - 4 + (y + 2)² - 4 = 0

(x - 2)² + (y + 2)² - 8 = 0

(x - 2)² + (y + 2)² = 8

7. Answer :

r = 2cos θ + 2sin θ

Multiply both sides by r.

r² = r(2cos θ + 2sin θ)

r² = 2rcos θ + 2rsin θ

Substitute x² + y² for r², x for r cos θ and y for r sin θ.

x² + y² = 2x + 2y

x² - 2x + y² - 2y = 0

x² - 2(x)(1) + y² - 2(y)(1) = 0

x² - 2(x)(1) + 1² - 1² + y² - 2(y)(1) + 1² - 1² = 0

(x - 1)² - 1² + (y - 1)² - 1² = 0

(x - 1)² - 1 + (y - 1)² - 1 = 0

(x - 1)² + (y - 1)² - 2 = 0

(x - 1)² + (y - 1)² = 2

8. Answer :

Substitute y for r sin θ and x for r sin θ.

9. Answer :

Multiply both sides by r.

Substitute x² + y² for r², x for r cos θ and y for r sin θ.

10. Answer :

r² = 5 sec (2θ)

r²cos (2θ) = 5

r²(cos² θ - sin² θ) = 5

r²cos² θ - r²sin² θ = 5

(rcos θ)² - (rsin θ)² = 5

Substitute x² + y² for r², x for r cos θ and y for r sin θ.

x² - y² = 5

11. Answer :

x = y²

Substitute r cos θ for x and r sin θ. for y.

r cos θ = (r sin θ)²

r cos θ = r² sin² θ

Divide both sides by r.

r sin² θ = cos θ

Divide both sides by sin² θ.

12. Answer :

y = x²

Substitute r sin θ for y and r cos θ for x.

r sin θ = (r cos θ)²

r sin θ = r² cos² θ

Divide both sides by r.

r cos² θ = sin θ

Divide both sides by cos² θ.

13. Answer :

2x - 5y = 4

Substitute r sin θ for y and r cos θ for x.

2(r cos θ) - 5(r sin θ) = 4

2r cos θ - 5r sin θ = 4

r(2cos θ - 5sin θ) = 4

Divide both sides by (2cos θ - 5sin θ).

14. Answer :

x² + y² = 9

Substitute r² for x² + y².

r² = 9

Taking square root on both sides,

r = ±3

15. Answer :

x² + y² - 7y = 0

Substitute r² for x² + y² and  r sin θ for y.

r² - 7(r sin θ) = 0

r² - 7rsin θ = 0

r² = 7rsin θ

Divide both sides by r.

r = 7sin θ

16. Answer :

(x - 1)² + (y + 1)² = 2

x² - 2x + 1 + y² + 2y + 1 = 2

x² + y² - 2x + 2y + 2 = 2

Subtract 2 from both sides.

x² + y² - 2x + 2y = 0

Substitute r² for x² + y², r cos θ for x and r sin θ for y.

r² - 2rcos θ + 2rsin θ = 0

r² = 2rcos θ - 2rsin θ

Divide both sides by r.

r = 2cos θ - 2sin θ

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