convert the polar equation to rectangular form.
1.) r sec(theta) = 3
2.) r = 4 cos(theta) - 4 sin(theta)
convert from rectangular equation to polar form.
1.) x^2 + (y-1)^2 = 1
2.) (x-1)^2 + (y+4)^2 = 17
To convert a polar equation to rectangular form, you can use some trigonometric identities. The general conversion formulas from polar to rectangular coordinates are:
x = r * cos(theta)
y = r * sin(theta)
1.) To convert the polar equation r sec(theta) = 3 to rectangular form:
First, we can rewrite sec(theta) as 1/cos(theta):
r * (1/cos(theta)) = 3
Next, substituting the formulas for x and y in terms of r and theta:
r * (1/x) = 3 (since cos(theta) = x/r)
Multiply both sides of the equation by x:
r = 3x
So, the rectangular form of the polar equation r sec(theta) = 3 is simply y = 3x.
2.) To convert the polar equation r = 4 cos(theta) - 4 sin(theta) to rectangular form:
Using the formulas x = r * cos(theta) and y = r * sin(theta), we substitute:
x = (4 cos(theta)) - (4 sin(theta))
y = (4 sin(theta)) + (4 cos(theta))
Simplifying these equations:
x = 4(cos(theta) - sin(theta))
y = 4(sin(theta) + cos(theta))
So, the rectangular form of the polar equation r = 4 cos(theta) - 4 sin(theta) is y = 4sin(theta) + 4cos(theta) or x - 4sin(theta) - 4cos(theta) = 0.
To convert a rectangular equation to polar form, you can use the formulas for r and theta:
r^2 = x^2 + y^2
tan(theta) = y/x
1.) To convert the rectangular equation x^2 + (y-1)^2 = 1 to polar form:
Using the formula for r^2:
r^2 = x^2 + y^2
Substituting the given equation:
r^2 = x^2 + (y-1)^2
Next, substituting the formula for r in terms of x and y:
(x^2 + (y-1)^2) = x^2 + y^2
Expanding and simplifying:
y^2 - 2y + 1 = 0
Solving this equation for y, we get:
y = 1
Substituting this back into the equation x^2 + (y-1)^2 = 1, we have x^2 + (1-1)^2 = 1^2, which simplifies to x^2 = 0.
Therefore, the polar form of the rectangular equation x^2 + (y-1)^2 = 1 is r = 0, which means the origin.
2.) To convert the rectangular equation (x-1)^2 + (y+4)^2 = 17 to polar form:
First, let's solve for x in terms of y:
x = y - 1
Substituting this into the given equation, we get:
(y - 1)^2 + (y + 4)^2 = 17
Expanding and simplifying:
2y^2 + 2y - 12 = 0
Factoring:
(y + 3)(y - 2) = 0
The solutions for y are y = -3 and y = 2.
For y = -3, we have x = -3 - 1 = -4.
For y = 2, we have x = 2 - 1 = 1.
So, the rectangular equation represents two points: (-4, -3) and (1, 2).
To convert these points to polar form, we can use the formulas for r and theta:
r = sqrt(x^2 + y^2)
tan(theta) = y/x
For the point (-4, -3):
r = sqrt((-4)^2 + (-3)^2) = 5
theta = atan((-3)/(-4)) = atan(3/4)
For the point (1, 2):
r = sqrt((1)^2 + (2)^2) = sqrt(5)
theta = atan(2/1) = atan(2)
Therefore, the polar form of the rectangular equation (x-1)^2 + (y+4)^2 = 17 is (r, theta) = (5, atan(3/4)) and (sqrt(5), atan(2)).