translate this polar equation into a rectangular form:
rsin(2theta)=sin(theta)
My answer:
r2sin(theta)cos(theta)=sin(theta)
2rcos(theta)=1
2x=1
x=1/2
This is the only answer I can get and someone told me it's wrong. Would someone please help me with this
or maybe the answer is:
y=2xy
but that's kind of weird
I don't know if this is right, but this is how I would work it.
rsin (2 theta) = sin (theta)
r (2 sin(theta) 2cos(theta)) = sin (theta) divide both sides by 2 sin
r 2 cos (theta) = 1/2 divide both sides by 2
r cos (theta) = 1/4
x = 1/4
To translate the given polar equation, rsin(2theta) = sin(theta), into rectangular form, you can start by using the double-angle identity for sine:
sin(2theta) = 2sin(theta)cos(theta)
Substituting this identity into the equation, we have:
r * 2sin(theta)cos(theta) = sin(theta)
Next, you can simplify by dividing both sides of the equation by sin(theta):
2rcos(theta) = 1
Now, to convert this equation into rectangular form, we need to substitute r and theta with their corresponding expressions:
r = sqrt(x^2 + y^2)
cos(theta) = x / sqrt(x^2 + y^2)
Using these substitutions, the equation becomes:
2(x / sqrt(x^2 + y^2)) = 1
Now, multiplying both sides by sqrt(x^2 + y^2), we get:
2x = sqrt(x^2 + y^2)
Squaring both sides of the equation, we have:
4x^2 = x^2 + y^2
Simplifying further:
3x^2 = y^2
This equation represents a parabola that opens either upwards or downwards depending on the sign of y^2. Therefore, the correct rectangular form of the polar equation rsin(2theta) = sin(theta) is 3x^2 - y^2 = 0.
I hope this helps! Let me know if you have any further questions.