Q: Which of the following is the correct process to convert $$r = a \cos \theta + b \sin \theta to $$rectangular form?

Multiply both sides by $$r$$, then substitute $$x = r \cos \theta$$ and $$y = r \sin \theta$$

Q: Given the rectangular equation $$x + 2y = 6$$, what is the equivalent polar equation?

$$r = 6 / (\cos \theta + 2 \sin \theta)$$

Q: A line in rectangular form is $$2x - y = 8. $$What is the equivalent polar equation?

$$r = 8 / (2 \cos \theta - \sin \theta)$$

Question 1

Which of the following equations represents a line in rectangular (Cartesian) form?

Accepted Answer

$$x + 2y = 5$$

Question 2

Given the polar equation $$r = 2 \cos 	heta + 3 \sin 	heta$$, which of the following is the correct rectangular form?

Accepted Answer

Question 3

What is the first step in converting the polar equation $$r = 4 \cos 	heta to $$rectangular form?

Accepted Answer

Multiply both sides by $$r$$

Question 4

Which substitution is used to convert $$r \cos 	heta to $$rectangular coordinates?

Accepted Answer

$$r \cos 	heta = x$$

Question 5

Convert the polar equation $$r = 5 to $$rectangular form.

Accepted Answer

$$x^2$$ + $$y^2 = 25$

Question 6

A line in polar form is given by $$r = 6 \sin 	heta. $$What is its rectangular form?

Accepted Answer

$$y = 6$$

Question 7

Which of the following is the correct process to convert $$r = a \cos 	heta + b \sin 	heta to $$rectangular form?

Accepted Answer

Multiply both sides by $$r$$, then substitute $$x = r \cos 	heta$$ and $$y = r \sin 	heta$$

Question 8

Given the rectangular equation $$x + 2y = 6$$, what is the equivalent polar equation?

Accepted Answer

$$r = 6 / (\cos 	heta + 2 \sin 	heta)$$

Question 9

If a circle is centered at the origin with radius $$r = 7$$, what is its equation in rectangular form?

Accepted Answer

$$x^2$$ + $$y^2 = 49$

Question 10

Which of the following best describes the geometric meaning of the equation $$r = a \cos 	heta + b \sin 	heta$$?

Accepted Answer

A straight line not passing through the origin

Question 11

A radar detects an object at a distance $$r = 10$$ units at an angle $$	heta = \frac{\pi}{3}. $$What are the rectangular coordinates $$(x, y) of $$the object?

Accepted Answer

$$(5, 5\sqrt{3})$$

Question 12

Which of the following is NOT a valid substitution when converting from polar to rectangular coordinates?

Accepted Answer

$$	heta = x / y$$

Question 13

Given $$r = 3 \cos 	heta$$, what is the rectangular equation?

Accepted Answer

$$x^2$$ + $$y^2 = 3x$

Question 14

A line in rectangular form is $$2x - y = 8. $$What is the equivalent polar equation?

Accepted Answer

$$r = 8 / (2 \cos 	heta - \sin 	heta)$$

Question 15

Which of the following real-world applications would most likely require converting between polar and rectangular coordinates?

Accepted Answer

Analyzing the trajectory of a satellite orbiting Earth

Solving Linear Equations: Rectangular and Polar Forms

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