1.Graph the polar equation r=3-2sin(theta)

2. Find the polar coordinates of 6 radical 3,6 for r > 0.

3. Find the rectangular coordinates of (7, 30°).

4. Write the rectangular equation in polar form.
(x – 4)2 + y2 = 16

5. Write the equation –2x + 6y = 7 in polar form.

6. Find the distance between P1(3, –195°) and P2(–4, –94°) on the polar plane. Round your answer to the nearest thousandth.

7. Write the polar equation in rectangular form.
r = –12 cos(theta)

1. To graph the polar equation r = 3 - 2sin(theta), you can follow these steps:

- Plot points for different values of theta.
- Substitute each value of theta into the equation to find the corresponding value of r.
- Convert the polar coordinates (r, theta) to rectangular coordinates (x, y) using the formulas x = r*cos(theta) and y = r*sin(theta).
- Plot the rectangular coordinates (x, y) on a Cartesian coordinate system.
- Connect the plotted points to form the graph of the polar equation.

2. To find the polar coordinates of (6√3, 6) for r > 0, we can use the formulas:
- r = sqrt(x^2 + y^2)
- theta = arctan(y/x)

Substitute the given rectangular coordinates into these formulas:
- r = sqrt((6√3)^2 + 6^2) = sqrt(108 + 36) = sqrt(144) = 12
- theta = arctan(6/6√3) = arctan(1/√3) = 30°

Therefore, the polar coordinates are (12, 30°).

3. To find the rectangular coordinates of (7, 30°), you can use the formulas:
- x = r*cos(theta)
- y = r*sin(theta)

Substitute the given polar coordinates into these formulas:
- x = 7*cos(30°) = 7*(√3/2) = 7√3/2
- y = 7*sin(30°) = 7*(1/2) = 7/2

Therefore, the rectangular coordinates are (7√3/2, 7/2).

4. To write the rectangular equation (x – 4)^2 + y^2 = 16 in polar form, you can use the formulas:
- x = r*cos(theta)
- y = r*sin(theta)

Substitute these formulas into the rectangular equation:
- (r*cos(theta) – 4)^2 + (r*sin(theta))^2 = 16
- r^2*cos^2(theta) – 8*r*cos(theta) + 16 + r^2*sin^2(theta) = 16
- r^2*(cos^2(theta) + sin^2(theta)) – 8*r*cos(theta) + 16 = 16
- r^2 – 8*r*cos(theta) = 0
- r*(r – 8*cos(theta)) = 0

Therefore, the polar equation is r = 8*cos(theta).

5. To write the equation –2x + 6y = 7 in polar form, you can use the formulas:
- x = r*cos(theta)
- y = r*sin(theta)

Substitute these formulas into the equation:
- –2*(r*cos(theta)) + 6*(r*sin(theta)) = 7
- –2r*cos(theta) + 6r*sin(theta) = 7

Therefore, the polar form of the equation is –2r*cos(theta) + 6r*sin(theta) = 7.

6. To find the distance between P1(3, –195°) and P2(–4, –94°) on the polar plane, you can use the distance formula:
- d = sqrt((r1*cos(theta1) - r2*cos(theta2))^2 + (r1*sin(theta1) - r2*sin(theta2))^2)

Substitute the given polar coordinates into this formula:
- d = sqrt((3*cos(-195°) - (-4)*cos(-94°))^2 + (3*sin(-195°) - (-4)*sin(-94°))^2)

Evaluate the trigonometric functions and calculate the distance.

7. To write the polar equation r = –12*cos(theta) in rectangular form, you can use the formulas:
- r = sqrt(x^2 + y^2)
- theta = arctan(y/x)

Substitute the given polar equation into these formulas:
- sqrt(x^2 + y^2) = -12*cos(theta)
- arctan(y/x) = theta

Square both sides of the first equation to eliminate the square root:
- x^2 + y^2 = 144*cos^2(theta)

Take the arctangent of both sides of the second equation to eliminate the theta:
- arctan(y/x) = theta

Therefore, the rectangular equation is x^2 + y^2 = 144*cos^2(theta) and arctan(y/x) = theta.