Polar coordinates - Math Central

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Question from kumi, a student:

Can you help me solve this problem?
2.Sketch the graph of r=(2)/(1-sinθ). Make sure you show a complete chart demonstrating how you arrive at your sketch, and sketch the graph neatly and clearly with proper labeling.
b. change the equation above to cartesian coordinates and identify the type of graph.
c. explain which coordinate system gives an easier function to graph and why
d.find a function of the form r=f(θ) for a circle centered at (8,π/2) and with the point (0,0) lying on the circle

Hi Kumi,

To sketch the graph I would choose values of θ where I know sinθ. For example θ = 0, ±π/4, ±π/2, π, ±3π/4, ±3π/2. For each find the value of r and then plot the values.

For part b. use the facts that r² = x² + y², x = r cosθ and y = r sinθ. Thus

r=2/(1-sinθ)

becomes

√(x² + y²) = 2/[1 - y/√(x² + y²)]

Take the denominator on the right side, write it to a common denominator and get

√(x² + y²) = [2 √(x² + y²)]/[√(x² + y²) - y]

which becomes

√(x² + y²) -y = 2

Move the y term to the right side, square both sides and simplify.

For part d. I would write the centre of the circle in Cartesian form, write the equation of the circle in Cartesian form and then convert it to polar form.

Penny

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