What is the polar form of the equation
3x - 4y = 12?
The answer is r = 12/ (3cosx - 4 sinx)
(x = theta)
I don't know how to get the answer though.
draw a right-angled triangle in the standard position in quadrant I with sides x,y and r and angle Ø
we know sinØ = y/r
so y = rsinØ
similarly
x = rcosØ
so just substitute...
3x - 4y = 12
3rcosØ - 4rsinØ = 12
r(3cosØ - 4sinØ) = 12
r = 12/(3cosØ - 4sinØ)
Wow, I never thought of factoring out the r, heh. Thanks.
To express the equation 3x - 4y = 12 in polar form, we can use the following steps:
Step 1: Convert the equation to Cartesian form.
To convert the equation into Cartesian form, we need to isolate y:
3x - 4y = 12
-4y = -3x + 12
y = (3/4)x - 3
Step 2: Substitute the Cartesian variables with their polar counterparts.
In polar form, we express x = r * cos(theta) and y = r * sin(theta). Substituting these values into the Cartesian equation, we get:
r * sin(theta) = (3/4)(r * cos(theta)) - 3
Step 3: Simplify the equation.
To simplify the equation further, distribute (3/4) to r * cos(theta):
r * sin(theta) = (3/4) * r * cos(theta) - 3
Step 4: Move all terms involving r to one side of the equation.
r * sin(theta) - (3/4) * r * cos(theta) = -3
Step 5: Factor out r from the left side of the equation.
r * (sin(theta) - (3/4) * cos(theta)) = -3
Step 6: Divide both sides of the equation by (sin(theta) - (3/4) * cos(theta)) to solve for r.
r = -3 / (sin(theta) - (3/4) * cos(theta))
Thus, the polar form of the equation 3x - 4y = 12 is r = -3 / (sin(theta) - (3/4) * cos(theta)).
To convert the equation 3x - 4y = 12 into polar form, we need to express it in terms of r and θ (where x = r cos θ and y = r sin θ). Here's how you can do it step by step:
1. Start by isolating x in terms of y:
3x = 4y + 12
x = (4y + 12)/3
2. Substitute x and y with their respective polar forms, x = r cos θ and y = r sin θ:
r cos θ = (4r sin θ + 12)/3
3. Multiply both sides of the equation by 3 to eliminate the denominator:
3r cos θ = 4r sin θ + 12
4. Rearrange the equation to express it in terms of r:
3r cos θ - 4r sin θ = 12
5. Factor out r on the left side:
r (3 cos θ - 4 sin θ) = 12
6. Finally, divide both sides by (3 cos θ - 4 sin θ):
r = 12 / (3 cos θ - 4 sin θ)
So, the polar form of the equation 3x - 4y = 12 is r = 12 / (3 cos θ - 4 sin θ), where x = r cos θ and y = r sin θ.