Convert the rectangular equation to polar form
3x-y+2=0
To convert the rectangular equation 3x - y + 2 = 0 to polar form, we can substitute x = r cos(theta) and y = r sin(theta), where r is the radius and theta is the angle in polar coordinates. This gives:
3(r cos(theta)) - (r sin(theta)) + 2 = 0
Simplifying the equation, we get:
r(3 cos(theta) - sin(theta)) = -2
Dividing both sides by the expression in the parentheses:
r = -2 / (3 cos(theta) - sin(theta))
This is the polar form of the equation.
Convert the rectangular equation to polar form
12.3x-y+2=0
13. xy=4
14. (x+y)-9(x-2)=0 72-36=0
15. y²-8x-16-0 (4-4)(3-4)=0
For the rectangular equation xy = 4, we can convert it to polar form using the substitution x = r cos(theta) and y = r sin(theta), which gives:
r cos(theta) * r sin(theta) = 4
r^2 sin(theta) cos(theta) = 4
Using the identity 2 sin(theta) cos(theta) = sin(2theta), we can simplify the equation to:
r^2 sin(2theta) = 8
r = sqrt(8 / sin(2theta))
This is the polar form of the equation.
For the rectangular equation (x+y) - 9(x-2) = 0, we can simplify it to:
x - 8y + 18 = 0
Then, we can convert it to polar form using the substitution x = r cos(theta) and y = r sin(theta), which gives:
r cos(theta) - 8r sin(theta) + 18 = 0
Simplifying the equation, we get:
r = 18 / (cos(theta) - 8sin(theta))
This is the polar form of the equation.
For the rectangular equation y^2 - 8x - 16 = 0, we can complete the square to get:
y^2 - 8x - 16 = (y - 0)^2 - 16 - 8x
Simplifying the equation, we get:
(y - 0)^2 = 8x + 16
Using the substitution x = r cos(theta) and y = r sin(theta), we get:
r^2 sin^2(theta) = 8r cos(theta) + 16
r^2 sin^2(theta) - 8r cos(theta) - 16 = 0
This equation does not simplify nicely into a standard form of polar equation, but it is still a valid polar form.
16. r = 4 sin θ
17.θ= (π/6)
18. r² = sin 2θ
19. r = 6/(2-3 sin θ)
To convert the polar equation r = 4 sin(theta) to rectangular form, we can use the following trigonometric identities:
sin(theta) = y / r
cos(theta) = x / r
Substituting these into the polar equation, we get:
r = 4 sin(theta)
r = 4 y / r
r^2 = 4 y
x^2 + y^2 = 4 y
This is the rectangular form of the equation.
The equation theta = pi/6 represents a line at an angle of pi/6 radians (30 degrees) from the positive x-axis in the polar coordinate system. In rectangular coordinates, this line has the equation y = x tan(pi/6) = x/sqrt(3).
To convert the polar equation r^2 = sin(2theta) to rectangular form, we can use the following trigonometric identities:
sin(2theta) = 2 sin(theta) cos(theta)
sin(theta) = y / r
cos(theta) = x / r
Substituting these into the polar equation, we get:
r^2 = sin(2theta)
r^2 = 2 sin(theta) cos(theta)
r^2 = 2 (y / r) (x / r)
x^2 + y^2 = 2xy
This is the rectangular form of the equation.
To convert the polar equation r = 6 / (2 - 3 sin(theta)) to rectangular form, we can first substitute sin(theta) = y / r and cos(theta) = x / r, giving:
r = 6 / (2 - 3y / r)
r(2 - 3y / r) = 6
2r - 3y = 6
This is the rectangular form of the equation.
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