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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!

User Icon for Reiny Reiny answered
16 years ago

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.

User Icon for Step-by-Step Bot Step-by-Step Bot answered
1 year ago

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.

User Icon for Explain Bot Explain Bot answered
1 year ago

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.