16. Parametric Equations & Polar Coordinates

Without calculating derivatives, determine the slopes of each of the lines tangent to the curve r=8 cos θ−4 at the origin.

Plot the point $(3,\(\frac{\pi}{2}\))$ & find another set of coordinates, $(r,θ)$, for this point, where:

(A) $r≥0,2π≤θ≤4π$,

(B) $r≥0,-2π≤θ≤0$,

(C) $r≤0,0≤θ≤2π$.

Plot the point $(-3,-\(\frac{\pi}{6}\))$, then identify which of the following sets of coordinates is the same point.

Plot the point on the polar coordinate system.

$(-2,\(\frac{2\pi}{3}\))$

Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

What is the polar equation of the horizontal line y = 5?

Identify whether the given equation is that of a cardioid, limaçon, rose, or lemniscate.

$r=1-\(\sin\]\theta\)$

Cartesian to Polar Coordinates

Find the polar coordinates, 0 ≤ θ ≤ 2π and r ≤ 0, of the following points given in Cartesian coordinates.

c. (−1, √3)

Convert each equation to its polar form.

$x^2+(y-2)^2=4$

Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).

b. Explain why tan θ = y/x.

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.

(-4, 3π/2)

Plot the point on the polar coordinate system.

$(5,210°)$

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.

r cos θ = -4

15–22. Sets in polar coordinates Sketch the following sets of points.

r = 3

Convert the point to polar coordinates.

$(1,1)$