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11.4E: Exercises

  • Page ID
    137821
    • Gilbert Strang & Edwin “Jed” Herman
    • OpenStax

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    In exercises 1 - 7, plot the point whose polar coordinates are given by first constructing the angle \(\theta\) and then marking off the distance \(r\) along the ray.

    1) \(\left(3,\frac{π}{6}\right)\)

    Answer
    On the polar coordinate plane, a ray is drawn from the origin marking π/6 and a point is drawn when this line crosses the circle with radius 3.

    2) \(\left(−2,\frac{5π}{3}\right)\)

    3) \(\left(0,\frac{7π}{6}\right)\)

    Answer
    On the polar coordinate plane, a ray is drawn from the origin marking 7π/6 and a point is drawn when this line crosses the circle with radius 0, that is, it marks the origin.

    4) \(\left(−4,\frac{3π}{4}\right)\)

    5) \(\left(1,\frac{π}{4}\right)\)

    Answer
    On the polar coordinate plane, a ray is drawn from the origin marking π/4 and a point is drawn when this line crosses the circle with radius 1.

    6) \(\left(2,\frac{5π}{6}\right)\)

    7) \(\left(1,\frac{π}{2}\right)\)

    Answer
    On the polar coordinate plane, a ray is drawn from the origin marking π/2 and a point is drawn when this line crosses the circle with radius 1.

    In exercises 8 - 11, consider the polar graph below. Give two sets of polar coordinates for each point.

    The polar coordinate plane is divided into 12 pies. Point A is drawn on the first circle on the first spoke above the θ = 0 line in the first quadrant. Point B is drawn in the fourth quadrant on the third circle and the second spoke below the θ = 0 line. Point C is drawn on the θ = π line on the third circle. Point D is drawn on the fourth circle on the first spoke below the θ = π line.

    8) Coordinates of point A.

    9) Coordinates of point B.

    Answer
    \(B\left(3,\frac{−π}{3}\right) B\left(−3,\frac{2π}{3}\right)\)

    10) Coordinates of point C.

    11) Coordinates of point D.

    Answer
    \(D\left(5,\frac{7π}{6}\right) D\left(−5,\frac{π}{6}\right)\)

    In exercises 12 - 17, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in \((0,2π]\). Round to three decimal places.

    12) \((2,2)\)

    13) \((3,−4)\)

    Answer
    \((5,−0.927),\;(−5,−0.927+π)\)

    14) \((8,15)\)

    15) \((−6,8)\)

    Answer
    \((10,−0.927),\;(−10,−0.927+π)\)

    16) \((4,3)\)

    17) \((3,−\sqrt{3})\)

    Answer
    \((2\sqrt{3},−0.524),\;(−2\sqrt{3},−0.524+π)\)

    In exercises 18 - 24, find rectangular coordinates for the given point in polar coordinates.

    18) \(\left(2,\frac{5π}{4}\right)\)

    19) \(\left(−2,\frac{π}{6}\right)\)

    Answer
    \((−\sqrt{3},−1)\)

    20) \(\left(5,\frac{π}{3}\right)\)

    21) \(\left(1,\frac{7π}{6}\right)\)

    Answer
    \(\left(−\frac{\sqrt{3}}{2},\frac{−1}{2}\right)\)

    22) \(\left(−3,\frac{3π}{4}\right)\)

    23) \(\left(0,\frac{π}{2}\right)\)

    Answer
    \((0,0)\)

    24) \((−4.5,6.5)\)

    In exercises 25 - 29, determine whether the graphs of the polar equation are symmetric with respect to the \(x\)-axis, the \(y\)-axis, or the origin.

    25) \(r=3\sin(2θ)\)

    Answer
    Symmetry with respect to the x-axis, y-axis, and origin.

    26) \(r^2=9\cos θ\)

    27) \(r=\cos\left(\frac{θ}{5}\right)\)

    Answer
    Symmetric with respect to x-axis only.

    28) \(r=2\sec θ\)

    29) \(r=1+\cos θ\)

    Answer
    Symmetry with respect to x-axis only.

    In exercises 30 - 33, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

    30) \(r=3\)

    31) \(θ=\frac{π}{4}\)

    Answer
    Line \(y=x\)

    32) \(r=\sec θ\)

    33) \(r=\csc θ\)

    Answer
    \(y=1\)

    In exercises 34 - 36, convert the rectangular equation to polar form and sketch its graph.

    34) \(x^2+y^2=16\)

    35) \(x^2−y^2=16\)

    Answer

    Hyperbola; polar form \(r^2\cos(2θ)=16\) or \(r^2=16\sec θ.\)

    A hyperbola with vertices at (−4, 0) and (4, 0), the first pointing out into quadrants II and III and the second pointing out into quadrants I and IV.

    36) \(x=8\)

    In exercises 37 - 38, convert the rectangular equation to polar form and sketch its graph.

    37) \(3x−y=2\)

    Answer

    \(r=\frac{2}{3\cos θ−\sin θ}\)

    A straight line with slope 3 and y intercept −2.

    38) \(y^2=4x\)

    In exercises 39 - 43, convert the polar equation to rectangular form and sketch its graph.

    39) \(r=4\sin θ\)

    40) \(x^2+y^2=4y\)

    Answer
    A circle of radius 2 with center at (2, π/2).

    41) \(r=6\cos θ\)

    42) \(r=θ\)

    Answer

    \(x\tan\sqrt{x^2+y^2}=y\)

    A spiral starting at the origin and crossing θ = π/2 between 1 and 2, θ = π between 3 and 4, θ = 3π/2 between 4 and 5, θ = 0 between 6 and 7, θ = π/2 between 7 and 8, and θ = π between 9 and 10.

    43) \(r=\cot θ\csc θ\)

    In exercises 44 - 54, sketch a graph of the polar equation and identify any symmetry.

    44) \(r=1+\sin θ\)

    Answer

    \(y\)-axis symmetry

    A cardioid with the upper heart part at the origin and the rest of the cardioid oriented up.

    45) \(r=3−2\cos θ\)

    46) \(r=2−2\sin θ\)

    Answer

    \(y\)-axis symmetry

    A cardioid with the upper heart part at the origin and the rest of the cardioid oriented down.

    47) \(r=5−4\sin θ\)

    48) \(r=3\cos(2θ)\)

    Answer

    \(x\)-and \(y\)-axis symmetry and symmetry about the pole

    A rose with four petals that reach their furthest extent from the origin at θ = 0, π/2, π, and 3π/2.

    49) \(r=3\sin(2θ)\)

    50) \(r=2\cos(3θ)\)

    Answer
    \(x\)-axis symmetry

    A rose with three petals that reach their furthest extent from the origin at θ = 0, 2π/3, and 4π/3.

    51) \(r=3\cos\left(\frac{θ}{2}\right)\)

    52) \(r^2=4\cos\left(\frac{2}{θ}\right)\)

    Answer

    \(x\)-and \(y\)-axis symmetry and symmetry about the pole

    The infinity symbol with the crossing point at the origin and with the furthest extent of the two petals being at θ = 0 and π.

    53) \(r^2=4\sin θ\)

    54) \(r=2θ\)

    Answer
    no symmetry
    A spiral that starts at the origin crossing the line θ = π/2 between 3 and 4, θ = π between 6 and 7, θ = 3π/2 between 9 and 10, θ = 0 between 12 and 13, θ = π/2 between 15 and 16, and θ = π between 18 and 19.

    55) [T] The graph of \(r=2\cos(2θ)\sec(θ).\) is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.

    56) [T] Use a graphing utility and sketch the graph of \(r=\dfrac{6}{2\sin θ−3\cos θ}\).

    Answer
    a line
    A line that crosses the y axis at roughly 3 and has slope roughly 3/2.

    57) [T] Use a graphing utility to graph \(r=\frac{1}{1−\cos θ}\).

    58) [T] Use technology to graph \(r=e^{\sin(θ)}−2\cos(4θ)\).

    Answer
    A geometric shape that resembles a butterfly with larger wings in the first and second quadrants, smaller wings in the third and fourth quadrants, a body along the θ = π/2 line and legs along the θ = 0 and π lines.

    59) [T] Use technology to plot \(r=\sin(\frac{3θ}{7})\) (use the interval \(0≤θ≤14π\)).

    60) Without using technology, sketch the polar curve \(θ=\frac{2π}{3}\).

    Answer
    A line with θ = 120°.

    61) [T] Use a graphing utility to plot \(r=θ\sin θ\) for \(−π≤θ≤π\).

    62) [T] Use technology to plot \(r=e^{−0.1θ}\) for \(−10≤θ≤10.\)

    Answer
    A spiral that starts in the third quadrant.

    63) [T] There is a curve known as the “Black Hole.” Use technology to plot \(r=e^{−0.01θ}\) for \(−100≤θ≤100\).

    64) [T] Use the results of the preceding two problems to explore the graphs of \(r=e^{−0.001θ}\) and \(r=e^{−0.0001θ}\) for \(|θ|>100\).

    Answer
    Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.

    This page titled 11.4E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.