convert from polar form into rectangular form. type the equation in rectangular form (simplified):
1) rsecθ = 3
2) r=5/cosθ-2sinθ
3) r=5cscθ
we know r^2 = x^2 + y^2 and x/r =cosØ, y/r = sinØ
given:
rsecØ = 3
r = 3/secØ
r = 3cosØ
r = 3(x/r)
r^2 = 3x
x^2 + y^2 = 3x , rewrite it any way you want.
the last one is done the same way
2nd :
I will assume you r = 5/(cosØ-2sinØ
r = 5/(x/r - 2y/r)
r = 5/((x-2y)/r)
r = 5 * r/(x - 2y)
1 = 5/(x-2y)
x-2y = 5
5. Find the Cartesian form of the equation, r3 = 3r cosØ.
To convert from polar form to rectangular form, we will use the following relationships:
r = √(x^2 + y^2)
θ = atan2(y, x)
1) rsecθ = 3:
To convert this equation to rectangular form, we need to express r in terms of x and y. Recall that secθ = 1/cosθ.
Using the relationship r = √(x^2 + y^2), we can substitute this into the equation:
√(x^2 + y^2) * (1/cosθ) = 3
Multiplying both sides by cosθ gives us:
√(x^2 + y^2) = 3cosθ
Now, square both sides of the equation:
(x^2 + y^2) = 9cos^2θ
Using the identity cos^2θ = 1 - sin^2θ, we can rewrite the equation:
(x^2 + y^2) = 9(1 - sin^2θ)
Simplifying further:
(x^2 + y^2) = 9 - 9sin^2θ
Now, use the relationship θ = atan2(y, x) to substitute sinθ into the equation:
(x^2 + y^2) = 9 - 9(sinθ)^2 = 9 - 9(y/r)^2 = 9 - 9y^2/r^2
Multiplying both sides by r^2:
r^2(x^2 + y^2) = 9r^2 - 9y^2
Since r = √(x^2 + y^2), we can substitute this into the equation:
r^3 = 9r^2 - 9y^2
This is the equation in rectangular form.
2) r = 5/cosθ - 2sinθ:
r = 5/cosθ - 2sinθ
To convert this equation to rectangular form, we will first express r in terms of x and y:
r = √(x^2 + y^2)
Substituting this into the equation:
√(x^2 + y^2) = 5/cosθ - 2sinθ
Now, we can square both sides of the equation to eliminate the square root:
x^2 + y^2 = (5/cosθ - 2sinθ)^2
Expanding and simplifying:
x^2 + y^2 = 25/cos^2θ - 20sinθ/cosθ + 4sin^2θ
Next, we can substitute sinθ and cosθ using the relationships:
sinθ = y/r
cosθ = x/r
x^2 + y^2 = 25/(x^2/r^2) - 20(y/r)/(x/r) + 4(y^2/r^2)
Multiplying both sides by r^2:
r^2(x^2 + y^2) = 25/r^2 - 20y/x + 4y^2
Simplifying:
x^2 + y^2 = 25/r^4 - 20y/x + 4y^2/r^2
This is the equation in rectangular form.
3) r = 5cscθ:
To convert this equation to rectangular form, we will first express r in terms of x and y:
r = √(x^2 + y^2)
Substituting this into the equation:
√(x^2 + y^2) = 5cscθ
Now, we can square both sides of the equation:
x^2 + y^2 = 25csc^2θ
Since cscθ = 1/sinθ, we can write the equation as:
x^2 + y^2 = 25/(sinθ)^2 = 25/(y/r)^2 = 25r^2/y^2
Multiplying both sides by y^2:
y^2(x^2 + y^2) = 25r^2
This is the equation in rectangular form.
To convert equations from polar form to rectangular form, we need to know the relationships between polar coordinates (r, θ) and rectangular coordinates (x, y).
1) Starting with rsecθ = 3:
To convert this equation into rectangular form, we need to express r and θ in terms of x and y.
The relationship between polar and rectangular coordinates is as follows:
x = r*cosθ
y = r*sinθ
From rsecθ = 3, we can rewrite it as:
r*(1/cosθ) = 3
Now, substitute r and θ with their respective rectangular forms:
(x^2 + y^2)*(1/x) = 3
Simplify the equation:
x^2 + y^2 = 3x
This is the simplified equation in rectangular form.
2) Moving to r = 5/(cosθ - 2sinθ):
We can use the same relationships between polar and rectangular coordinates.
Start by rearranging the equation:
rcosθ - 2rsinθ = 5
Substitute r and θ with their respective rectangular forms:
(x*cosθ + y*sinθ) - 2(x*sinθ - y*cosθ) = 5
Expand the equation:
x*cosθ + y*sinθ - 2x*sinθ + 2y*cosθ = 5
Rearrange the terms:
x*(cosθ - 2*sinθ) + y*(sinθ + 2*cosθ) = 5
This is the simplified equation in rectangular form.
3) Finally, considering r = 5cscθ:
Similar to before, we substitute r and θ with their respective rectangular forms:
(x^2 + y^2)(1/y) = 5
Simplify:
x^2 + y^2 = 5y
This is the simplified equation in rectangular form.
Note: In some cases, it may not be possible to simplify the equations further due to the nature of the functions involved.