The option that depicts the polar equation r = (cot 2θ)(csc θ) as an equation that is rectangular is: y² = 2x.
A polar expression is any equation that describes the correlation between r and θ, where r signifies the anticlockwise angle formed by a point on a curve, the pole, and the positive x-axis, and means the breadth between the origin and a location on a curve.
Step 1 - expand the polar equation
r = 2cosθ * cscθ
Step 2 - convert csc and cos to their equivalents in tan and sin
r = 2 * (1/tanθ) * (1/sinθ) * sinθ
r = 2 * (1/tanθ) * sinθ/sinθ
r = 2 * 1/tanθ
r = 2/tanθ, making 2 the subject of the formula, we have
rtanθ = 2
Note that rectangular coordinates of a point will be depicted as (x,y) and it's plolar coordinate will be (r, θ).
This mean
x = r cos θ; and
y = r sinθ
From the above, we can state that:
Since y = r sinθ
⇒ tanθ = y/x
⇒ (rsinθ/y) * tanθ = 2
Thus y * (y/x) = 2
Therefore,
y²/x = 2
y² = 2x
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