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(θ), you can plot multiple points with different values of θ and r, and connect them to form a curve. Here's how you can do it:
- Choose values for θ and calculate the corresponding r using the equation.
- Plot the obtained points on a polar coordinate system.
- Connect the points to form a smooth curve.
For example, let's choose values of θ from 0 to 2π (a full circle) with a small increment, say 0.1. Substitute these values into the equation and calculate r. Then plot the points (r, θ) on a polar graph. After connecting them, you will have the graph of the polar equation r = 3 - 2sin(θ).
2. To find the polar coordinates of a point (6√3, 6) given r > 0, you can use the formula:
r = √(x^2 + y^2)
θ = arctan(y / x)
Substituting the given values (x = 6√3, y = 6), we have:
r = √((6√3)^2 + 6^2) = √(108 + 36) = √144 = 12
θ = arctan(6 / (6√3)) = arctan(1/√3) = 30°
Therefore, the polar coordinates are (r, θ) = (12, 30°).
3. To find the rectangular coordinates of a point given in polar form (r, θ), you can use the following formulas:
x = r * cos(θ)
y = r * sin(θ)
Substituting the given values (r = 7, θ = 30°), we have:
x = 7 * cos(30°) = 7 * √(3) / 2 = 7√3 / 2
y = 7 * sin(30°) = 7 * 1 / 2 = 7 / 2
Therefore, the rectangular coordinates are (x, y) = (7√3 / 2, 7/2).
4. To write the rectangular equation (x – 4)^2 + y^2 = 16 in polar form, you can use the following transformations:
x = r * cos(θ)
y = r * sin(θ)
Substituting these into the equation, we have:
(r * cos(θ) – 4)^2 + (r * sin(θ))^2 = 16
r^2 * cos^2(θ) – 8r * cos(θ) + 16 + r^2 * sin^2(θ) = 16
r^2 * (cos^2(θ) + sin^2(θ)) – 8r * cos(θ) + 16 = 16
r^2 – 8r * cos(θ) = 0
Therefore, the polar form of the equation is r = 8 * cos(θ).
5. To write the equation –2x + 6y = 7 in polar form, you can use the following transformations:
x = r * cos(θ)
y = r * sin(θ)
Substituting these into the equation, we have:
–2(r * cos(θ)) + 6(r * sin(θ)) = 7
–2r * cos(θ) + 6r * sin(θ) = 7
This equation cannot be simplified further in polar form, so it remains as –2r * cos(θ) + 6r * sin(θ) = 7.
6. To find the distance between two points P1(3, –195°) and P2(–4, –94°) on the polar plane, you can use the formula:
d = √((r2^2 + r1^2) - 2*r1*r2*cos(θ2 - θ1))
Substituting the given values, we have:
r1 = 3, θ1 = –195°
r2 = –4, θ2 = –94°
Converting the angles to radians:
θ1 = –195° * π / 180 = –3.403
θ2 = –94° * π / 180 = –1.640
Using the formula, we get:
d = √(((-4)^2 + 3^2) - 2*3*(-4)*cos(-1.640 - (-3.403)))
= √(16 + 9 + 24cos(1.763))
≈ √(25 + 24cos(1.763))
Calculating the value of cos(1.763) and substituting it into the equation, you can find the distance d.
7. To write the polar equation r = –12cos(θ) in rectangular form, you can use the following transformation:
r = √(x^2 + y^2)
Substituting this into the equation, we have:
√(x^2 + y^2) = –12cos(θ)
Squaring both sides, we get:
x^2 + y^2 = 144cos^2(θ)
Therefore, the rectangular equation in terms of x and y is x^2 + y^2 = 144cos^2(θ).