Chapter 9: Q 31. (page 730)

In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.
$(3, - 3)$

Short Answer

Expert verified

The polar coordinates are $(3 \sqrt{2}, - \frac{π}{4} + 2 kπ) and (- 3 \sqrt{2}, \frac{3 π}{4} + 2 kπ)$ , for any integer k.

Step by step solution

Step 1. Given information.

The given rectangular coordinate is:

$(3, - 3)$

Here $x = 3 and y = - 3$ .

Step 2. Find the value of r.

To find the value of r, use the formula $r = \sqrt{x^{2} + y^{2}}$ :

$r = \sqrt{{(3)}^{2} + {(- 3)}^{2}} r = \sqrt{9 + 9} r = \sqrt{18} r = \pm 3 \sqrt{2}$

Step 3. Find the value of θ.

Use the formula $\tan θ = \frac{y}{x}$ ,

$\tan θ = \frac{y}{x}; then θ = \tan^{- 1} (\frac{y}{x}) θ = \tan^{- 1} (\frac{- 3}{3}) θ = \tan^{- 1} (- 1) θ = - \frac{π}{4}, \frac{π}{4}$

Step 4. Find the polar coordinates.

Take $r = 3 \sqrt{2}, θ = - \frac{π}{4}; then (r, θ) = (3 \sqrt{2}, - \frac{π}{4})$ ,

The coordinates of $(r, θ) are (3 \sqrt{2}, - \frac{π}{4} + 2 kπ)$ , for any integer k.

Now take $r = - 3 \sqrt{2}, θ = \frac{3 π}{4}; then (r, θ) = (- 3 \sqrt{2}, \frac{3 π}{4})$ ,

The coordinates of $(r, θ) are (- 3 \sqrt{2}, \frac{3 π}{4} + 2 kπ)$ , for any integer k.

Therefore, all the polar coordinates are $(3 \sqrt{2}, - \frac{π}{4} + 2 kπ) and (- 3 \sqrt{2}, \frac{3 π}{4} + 2 kπ)$ , for any integer k.

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In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.
$(3, - 3)$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the value of r.

Step 3. Find the value of θ.

Step 4. Find the polar coordinates.

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In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.3,-3

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the value of r.

Step 3. Find the value of θ.

Step 4. Find the polar coordinates.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Pure Maths

Mechanics Maths

Decision Maths

Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises 24–31 find all polar coordinate representations for the point given in rectangular coordinates.
$(3, - 3)$