10.3E: Exercises for Section 10.3

In exercises 1 - 7, plot the following polar coordinates.

1) \(\left(3,\frac{π}{6}\right)\)

Answer

2) \(\left(−1,\frac{5π}{4}\right)\)

Answer

3) \(\left(0,\frac{7π}{6}\right)\)

Answer

4) \(\left(−3, -5π/6 \right)\)

Answer

5) \(\left(1,\frac{π}{4}\right)\)

Answer

6) \(\left(-3,\frac{7π}{6}\right)\)

Answer

7) \(\left(1,\frac{π}{2}\right)\)

Answer

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

8) Coordinates of point A.

Answer

Point A: \(\left(1,\frac{π}{6}\right) \) or \(\left(−1,\frac{7π}{6}\right)\)

9) Coordinates of point B.

Answer

Point B: \(\left(3,\frac{−π}{3}\right)\) or \(\left(−3,\frac{2π}{3}\right)\)

10) Coordinates of point C.

Answer

Point C: \(\left(3,−π \right)\) or \(\left(−3,0 \right)\)

11) Coordinates of point D.

Answer

Point D: \(\left(5,\frac{7π}{6}\right) \) or \(\left(−5,\frac{π}{6}\right)\)

In exercises 12 - 17, the rectangular coordinates of a point are given. Find its polar coordinates with \(r>0\) and \(0 \leq \theta <2\pi\).

12) \((2,2)\)

Answer

\( \big( 2\sqrt{2}, \, \pi/4 \big) \)

13) \((0,−4)\)

Answer

\( \big( 4, \, 3\pi/2 \big) \)

14) \((-6,6\sqrt{3})\)

Answer

\( \big( 12, \, 2\pi/3 \big) \)

15) \((\sqrt{2}, -\sqrt{2})\)

Answer

\( \big( 2, \, 7\pi/4 \big) \)

16) \((-6,0)\)

Answer

\( \big( 6, \, \pi \big) \)

17) \(( -3\sqrt{3}, -3) \)

Answer

\( \big( 6, \, 7\pi/6 \big) \)

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

18) \(\left(2,\frac{5π}{4}\right)\)

Answer

\( \big( -\sqrt{2} , \, -\sqrt{2} \big) \)

19) \(\left(−2,\frac{π}{6}\right)\)

Answer

\((−\sqrt{3}, \, −1)\)

20) \(\left(-6,\frac{5π}{3}\right)\)

Answer

\( \big( -3, \, 3\sqrt{3} \big) \)

21) \(\left(1,\frac{7π}{6}\right)\)

Answer

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

22) \(\left(−8\sqrt{2}, -\frac{3π}{4}\right)\)

Answer

\( \big( 8 , \, 8 \big) \)

23) \(\left(0,\frac{π}{2}\right)\)

Answer

\((0,0)\)

24) \((5,2)\)

Answer

Answer rounded to the nearest thousandths is \((-2.081, \, 4.565)\).

In exercises 25-29, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

25) \(r=3\)

Answer

Circle \(x^2+y^2=9\)

26) \(θ=\frac{4π}{3}\)

Answer

The line \(y=x\sqrt{3}\)

27) \(r=4\sec θ\)

Answer

Line \(x=4\)

28) \(r=-\csc θ\)

Answer

\(y=-1\)

29) \(\displaystyle r=\frac{4}{\cos \theta + 2\sin \theta}\)

Answer

The line \(x+2y=4\)

In exercises 30 - 34, convert the given rectangular equation to polar form. Sketch its graph.

30) \(x^2+y^2=10\)

Answer

Circle \( r=\sqrt{10} \)

31) \(x^2−y^2=16\)

Answer

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

32) \(x=8\)

Answer

\(r=8\sec \theta\). Graph is a vertical line.

33) \(3x−y=2\)

Answer

\(r=\frac{2}{3\cos θ−\sin θ}\)

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

Answer

\(r=4\cot θ \csc θ}\). Graph is a parabola.

In the following exercises, sketch the graph of the following polar equations.

35) \(r=4\cos θ\)

Answer

The rectangular form is \((x-2)^2+y^2=4\). Graph is a circle.

36) \(r=4\sin \theta\)

Answer

The rectangular form is \(x^2+y^2=4y\).

37) \(r=θ\)

Answer

38) \(r=-\tan θ\sec θ\)

Answer

The rectangular form is \(y=-x^2\)

39) \(r=1+\sin θ\)

Answer

40) \(r=3−2\cos θ\)

Answer

41) \(r=2−2\sin θ\)

Answer

42) \(r=2−4\sin θ\)

Answer

43) \(r=3\cos(2θ)\)

Answer

44) \(r=2\cos(3θ)\)

Answer

\(x\)-axis symmetry

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