Find a cartesian equation for the curve described by the given polar equation.
a. r=2
b. r=3sin pheta
c.r^2=sin2pheta
I don't understand how to solve for this, especially for r squared. Would someone plz explain how o convert to a cartesian equation. any help would be greatly appreciated!
The relationships you need should be in your text, or you can find them at the top of
http://mathworld.wolfram.com/PolarCoordinates.html
your first one r = 2 is then quite easy
r = √(x^2 + y^2)
2 = √(x^2 + y^2)
x^2 + y^2 = 4 which is a circle
your second:
r=3sin pheta
r=3sinß
but sinß = y/r
r=3sinß
r = 3(y/r)
r^2 = 3y
x^2 + y^2 = 3y
for the third I can't tell if you mean
r^2 = sin(2ß) or r^2 = sin2ß
I will let you decide and then follow my previous examples.
To convert a polar equation to a Cartesian equation, you can use the following relationships:
x = r * cos(θ)
y = r * sin(θ)
Let's solve each of the given polar equations step-by-step:
a. r = 2
To convert this to a Cartesian equation, substitute the values of x and y using the relationships mentioned earlier:
x = 2 * cos(θ)
y = 2 * sin(θ)
Hence, the Cartesian equation is x = 2 * cos(θ) and y = 2 * sin(θ).
b. r = 3sin(θ)
Similarly, substitute the values of x and y using the relationships mentioned earlier:
x = 3sin(θ) * cos(θ)
y = 3sin(θ) * sin(θ)
You can simplify further:
x = 3sin(θ) * cos(θ) = 3/2 * 2sin(θ) * cos(θ) = 3/2 * sin(2θ)
y = 3sin^2(θ)
The Cartesian equation is x = 3/2 * sin(2θ) and y = 3sin^2(θ).
c. r^2 = sin(2θ)
To convert r^2, square both sides of the equation:
r^2 = sin(2θ)
(r * cos(θ))^2 + (r * sin(θ))^2 = sin(2θ)
Expand and simplify the equation:
r^2 * cos^2(θ) + r^2 * sin^2(θ) = sin(2θ)
x^2 + y^2 = sin(2θ)
The Cartesian equation is x^2 + y^2 = sin(2θ).
I hope this explanation helps! Let me know if you have any further questions.
To convert a polar equation to a Cartesian equation, we can make use of the relationships between polar and Cartesian coordinates. The conversion can be done using the following trigonometric identities
x = r·cos(θ)
y = r·sin(θ)
Let's apply these conversions to the given polar equations:
a. r = 2:
To convert this equation, we can substitute x = r·cos(θ) and y = r·sin(θ):
x = 2·cos(θ)
y = 2·sin(θ)
Therefore, the Cartesian equation for r = 2 is x^2 + y^2 = 4.
b. r = 3sin(θ):
Substituting x = r·cos(θ) and y = r·sin(θ) in the equation:
x = 3sin(θ)·cos(θ)
y = 3sin^2(θ)
Using trigonometric identities, we can rewrite:
2y = 3sin(2θ)
This is the Cartesian equation for r = 3sin(θ).
c. r^2 = sin(2θ):
Square both sides of the equation:
r^2 = sin(2θ)
(r·cos(θ))^2 + (r·sin(θ))^2 = sin(2θ)
Using trigonometric identities, we can simplify further:
x^2 + y^2 = sin(2θ)
So, the Cartesian equation for r^2 = sin(2θ) is x^2 + y^2 = sin(2θ).
By applying the respective substitutions and simplifications, we have obtained the Cartesian equations for the given polar equations.