Can you please check and correct my answers?
Convert to Rectangular: r*tanΘ/secΘ=2
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y=2
y=½
x=2
x=½
answer: y=2
Given: 5cos6Θ
What is the shape of the function?
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Limacon
Rose
Lemniscate
Circle
answer: rose
Given: 5cos6Θ
How many petals will this function have?
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5
6
12
None, this is not the case
answer: 12
Given: r = 4/-2-6sintheta
What is the eccentricity of the function?
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2
-2
3
-3
answer:-2
Given: r = 4/-2-6sintheta
What is the distance between the pole and the directrix?
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2
2/3
3
6
answer:6
Given: r = 4/-2-6sintheta
What type of directrix does this conic have?
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Vertical
Horizontal
Oblique
This conic does not have a directrix.
answer: horizontal
Given: r = 4/-2-costheta
What is the eccentricity of the function?
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½
-2
2
-½
answer:2
Given: r = 4/-2-costheta
What type of conic does this represent?
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Circle
Ellipse
Hyperbola
Parabola
answer:ellipse
Given: r = 4/-2-costheta
What is the distance between the pole and the directrix?
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4
-4
½
-½
answer:4
Given: r = 4/-2-costheta
What type of directrix does this conic have?
________________________________________
Vertical
Horizontal
Oblique
This conic does not have a directrix.
answer:vertical
For the first question:
To convert to rectangular form, you can use the relationships between the trigonometric functions and the coordinate variables in polar coordinates.
Given: r*tanΘ/secΘ = 2
We need to replace r and Θ with their corresponding expressions in rectangular form:
r = sqrt(x^2 + y^2)
Θ = arctan(y/x)
Replacing r and Θ in the equation:
(sqrt(x^2 + y^2) * tan(arctan(y/x)) / sec(arctan(y/x)) = 2
Simplifying further:
(sqrt(x^2 + y^2) * y/x * cos(arctan(y/x)) = 2
Taking the right-hand side expression:
2 = 2
This equation is true for any value of x and y. Thus, there are infinitely many solutions for x and y that satisfy this equation.
So, there is no unique rectangular form for this equation. Therefore, it cannot be converted to rectangular form.