Q: Which of the following is the correct conversion from polar to Cartesian coordinates?

$$x = r\cos\theta,\ y = r\sin\theta$$

Question 1

Which of the following is the correct polar equation for a circle centered at the origin with radius 3?

Accepted Answer

$$r = 3$$

Question 2

Given the polar equation $$r = 2\cos	heta$$, what is the Cartesian equation of the curve?

Accepted Answer

$$x^2$$ + $$y^2 = 2x$

Question 3

A point in polar coordinates is given as $$(r, 	heta) = (4, \frac{\pi}{3}). $$What are its Cartesian coordinates $$(x, y)$$?

Accepted Answer

$$(2, 2\sqrt{3})$$

Question 4

Which of the following describes the graph of $$r = 2 in $$polar coordinates?

Accepted Answer

A circle centered at the origin with radius 2

Question 5

Convert the Cartesian equation $$x^2$$ + $$y^2 = 9$ to polar form.

Accepted Answer

$$r = 3$$

Question 6

If $$r = 4\sin	heta$$, what is the Cartesian equation of the curve?

Accepted Answer

$$x^2$$ + $$y^2 = 4y$

Question 7

A medical imaging device rotates around a patient, collecting data at every $$30^\circ. If $$the device starts at $$(r, 	heta) = (5, 0)$$, what is the next data point in polar coordinates?

Accepted Answer

$$(5, \frac{\pi}{6})$$

Question 8

Which of the following is true about the graph of $$r = 2\cos	heta$$?

Accepted Answer

It is a circle tangent to the origin

Question 9

If $$r = 3\sin	heta$$, what is the center and radius of the corresponding circle in Cartesian coordinates?

Accepted Answer

Center $$(0, \frac{3}{2})$$, radius $$\frac{3}{2}$$

Question 10

A radar system detects an object at $$(r, 	heta) = (6, \frac{\pi}{4}). $$What is the $$x-$$coordinate of the object?

Accepted Answer

$$6\cos\frac{\pi}{4} = 3\sqrt{2}$$

Question 11

Which of the following is the correct conversion from polar to Cartesian coordinates?

Accepted Answer

$$x = r\cos	heta,\ y = r\sin	heta$$

Question 12

A lighthouse is located at the origin. A ship is detected at $$(r, 	heta) = (10, \frac{\pi}{6}). $$What is the distance of the ship from the $$y-$$axis?

Accepted Answer

$$10\cos\frac{\pi}{6} = 5\sqrt{3}$$

Question 13

Which of the following polar equations represents a circle that does NOT pass through the origin?

Accepted Answer

$$r = 2\cos	heta + 3$$

Question 14

If $$r = 5$$, what is the area enclosed by the curve?

Accepted Answer

$$25\pi$$

Question 15

A weather balloon is tracked at $$(r, 	heta) = (8, \frac{\pi}{3}). $$What is its $$y-$$coordinate?

Accepted Answer

$$8\sin\frac{\pi}{3} = 4\sqrt{3}$$

Question 16

Which of the following is the correct formula for converting $$x^2$$ + $$y^2$$ = $$r^2 to $$polar coordinates?

Accepted Answer

$$r = 	ext{constant}$$

Graphing and Converting Polar Equations in Trigonometry

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