Find a Cartesian equation for the curve and identify it.
r=5cscx
r = 5cscØ
r = 5/sinØ
we know that y/r = sinØ
or r = y/sinØ
then y/sinØ = 5/sinØ
y = 5 , a horizontal line
check:
http://www.wolframalpha.com/input/?i=polar+plot+r%3D5cscx
r = 5 cscθ
r sinθ = 5
y = 5
To find the Cartesian equation for the curve r = 5csc(x), we can convert the polar equation to rectangular coordinates using the following formulas:
x = r * cos(theta)
y = r * sin(theta)
Using the given equation r = 5csc(x), we have:
x = 5csc(x) * cos(theta)
y = 5csc(x) * sin(theta)
To further simplify, we can express csc(x) in terms of sine:
x = 5 * (1/sin(x)) * cos(theta)
y = 5 * (1/sin(x)) * sin(theta)
Using reciprocal identities, we can rewrite the equations as:
x = 5 * cot(x) * cos(theta)
y = 5 * cot(x) * sin(theta)
So, the Cartesian equation for the curve r = 5csc(x) is:
x = 5 * cot(x) * cos(theta)
y = 5 * cot(x) * sin(theta)
The curve does not have a specific name but is expressed in terms of its polar equation.
To find the Cartesian equation for the curve defined by the polar equation r = 5csc(x), we need to convert the equation from polar coordinates to rectangular (Cartesian) coordinates.
The first step is to express the trigonometric function of csc(x) in terms of sine and cosine. Recall that csc(x) = 1/sin(x). Substituting this into the equation, we have:
r = 5 / sin(x)
Next, we need to use the relationships between polar and Cartesian coordinates. Recall that x = r * cos(theta) and y = r * sin(theta). Rearranging the equation r = 5 / sin(x), we get:
sin(x) = 5 / r
Since r = sqrt(x^2 + y^2), we can substitute this into the equation:
sin(x) = 5 / sqrt(x^2 + y^2)
Now, using the relationship y = r * sin(theta), we can substitute y = r * sin(x):
sin(x) = 5 / sqrt(x^2 + (r * sin(x))^2)
Simplifying further:
sin(x) = 5 / sqrt(x^2 + (ysin(x))^2)
sin(x) = 5 / sqrt(x^2 + (ysin(x))^2)
sin(x)^2 = 25 / (x^2 + y^2 * sin(x)^2)
1 - cos(x)^2 = 25 / (x^2 + y^2 - y^2 * cos(x)^2)
(x^2 + y^2 - y^2 * cos(x)^2) - 25 cos(x)^2 = 0
x^2 + y^2 - (y^2 + 25)cos(x)^2 = 0
x^2 + y^2 - (y^2 + 25)(1 - sin(x)^2) = 0
x^2 + y^2 - (y^2 + 25)(1 - y^2/r^2) = 0
x^2 + y^2 - (y^2 + 25)(1 - y^2/ (x^2 + y^2)) = 0
x^2 + y^2 - y^2 - 25 + 25y^2/ (x^2 + y^2) = 0
x^2 - 25 + 24y^2/ (x^2 + y^2) = 0
Therefore, the Cartesian equation for the curve r = 5csc(x) is:
x^2 - 25 + 24y^2/ (x^2 + y^2) = 0
This equation represents an ellipse with its center at the origin (0, 0).