Problems

01

Convert points: Cartesian to Polar

Convert the Cartesian (rectangular) coordinates for these points into polar coordinates:

(a) (b) (c) (d)

Solution

01

(a)

Q1, SAFE, .

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(b)

Q1, SAFE, .

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(c)

Q2, UNSAFE, and we add to this angle.

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(d)

Q2, UNSAFE, (use 30-60-90 triangle) and we add to this angle.

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02

Convert equations: Polar to Cartesian

Convert the polar equation to a Cartesian equation. Be sure to simplify.

(a) (b) (c)

Solution

02

(a)

---

(b)

Use and therefore:

So:

This is a circle centered at with radius .

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(c)

Using and :

Note: This assumes that .

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03

Convert points: Polar to Cartesian

Convert the polar coordinates for these points into Cartesian (rectangular) coordinates:

(a) (b) (c) (d)

Solution

04

(a)

---

(b)

Method 1:

First compute and . Then and .

Method 2:

Observe . Then , and obtain the answer.

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(c)

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(d)

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04

Convert equations: Cartesian to Polar

Convert the Cartesian equation to a polar equation. Be sure to simplify.

(a) (b) (c)

Solution

05

(a) (b) (c)

(a)

Insert and :

---

(b)

---

(c)

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05

Sketching limaçons

Sketch the graphs of the following polar functions:

(a) (b)

(c) (d)

Solution

06

(a)

(b)

(c)

(d)

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06

Sketching roses

Sketch the graphs of the following polar functions. Use numbers to label the order in which the leaves/loops are traversed.

(a) (b) (c)

Solution

07

(a)

Numbers should be placed on the loops, 1, 2, 3, 4, starting in Q1 and going clockwise.

(b)

Numbers should be placed on the loops, 1, 2, 3, starting in Q1 and going clockwise.

(c)

Numbers should be placed on the loops, 1, 2, 3, 4, starting on the -axis and going clockwise.

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