Understanding Polar Coordinates

In the following exercises (1-4), plot the point whose polar coordinates are given by first constructing the angle [latex]\theta[/latex] and then marking off the distance [latex]r[/latex] along the ray.

1. [latex]\left(3,\dfrac{\pi }{6}\right)[/latex]
2. [latex]\left(0,\dfrac{7\pi }{6}\right)[/latex]
3. [latex]\left(1,\dfrac{\pi }{4}\right)[/latex]
4. [latex]\left(1,\dfrac{\pi }{2}\right)[/latex]

For the following exercises (5-6), consider the polar graph below. Give two sets of polar coordinates for each point.

5. Coordinates of point B.
6. Coordinates of point D.

For the following exercises (7-9), the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in [latex]\left(0,2\pi \right][/latex]. Round to three decimal places.

7. [latex]\left(3,-4\right)[/latex]
8. [latex]\left(-6,8\right)[/latex]
9. [latex]\left(3,\text{-}\sqrt{3}\right)[/latex]

For the following exercises (10-12), find rectangular coordinates for the given point in polar coordinates.

10. [latex]\left(-2,\dfrac{\pi }{6}\right)[/latex]
11. [latex]\left(1,\dfrac{7\pi }{6}\right)[/latex]
12. [latex]\left(0,\dfrac{\pi }{2}\right)[/latex]

For the following exercises (13-15), determine whether the graphs of the polar equation are symmetric with respect to the [latex]x[/latex] -axis, the [latex]y[/latex] -axis, or the origin.

13. [latex]r=3\sin\left(2\theta \right)[/latex]
14. [latex]r=\cos\left(\dfrac{\theta }{5}\right)[/latex]
15. [latex]r=1+\cos\theta[/latex]

For the following exercises (16-17), describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

16. [latex]\theta =\dfrac{\pi }{4}[/latex]
17. [latex]r=\csc\theta[/latex]

For the following exercise, convert the rectangular equation to polar form and sketch its graph.

18. [latex]{x}^{2}-{y}^{2}=16[/latex]

For the following exercise, convert the rectangular equation to polar form and sketch its graph.

19. [latex]3x-y=2[/latex]

For the following exercises (20-21), convert the polar equation to rectangular form and sketch its graph.

20. [latex]r=4\sin\theta[/latex]
21. [latex]r=\theta[/latex]

For the following exercises (22-27), sketch a graph of the polar equation and identify any symmetry.

22. [latex]r=1+\sin\theta[/latex]
23. [latex]r=2 - 2\sin\theta[/latex]
24. [latex]r=3\cos\left(2\theta \right)[/latex]
25. [latex]r=2\cos\left(3\theta \right)[/latex]
26. [latex]{r}^{2}=4\cos\left(2\theta \right)[/latex]
27. [latex]r=2\theta[/latex]

Area and Arc Length in Polar Coordinates

For the following exercises (1-6), determine a definite integral that represents the area.

1. Region enclosed by [latex]r=3\sin\theta[/latex]
2. Region enclosed by one petal of [latex]r=8\sin\left(2\theta \right)[/latex]
3. Region below the polar axis and enclosed by [latex]r=1-\sin\theta[/latex]
4. Region enclosed by the inner loop of [latex]r=2 - 3\sin\theta[/latex]
5. Region enclosed by [latex]r=1 - 2\cos\theta[/latex] and outside the inner loop
6. Region common to [latex]r=2\text{ and }r=4\cos\theta[/latex]

For the following exercises (7-13), find the area of the described region.

7. Enclosed by [latex]r=6\sin\theta[/latex]
8. Below the polar axis and enclosed by [latex]r=2-\cos\theta[/latex]
9. Enclosed by one petal of [latex]r=3\cos\left(2\theta \right)[/latex]
10. Enclosed by the inner loop of [latex]r=3+6\cos\theta[/latex]
11. Common interior of [latex]r=4\sin\left(2\theta \right)\text{and }r=2[/latex]
12. Common interior of [latex]r=6\sin\theta \text{ and }r=3[/latex]
13. Common interior of [latex]r=2+2\cos\theta \text{ and }r=2\sin\theta[/latex]

For the following exercises (14-15), find a definite integral that represents the arc length.

14. [latex]r=1+\sin\theta[/latex] on the interval [latex]0\le \theta \le 2\pi[/latex]
15. [latex]r={e}^{\theta }\text{ on the interval }0\le \theta \le 1[/latex]

For the following exercises (16-17), find the length of the curve over the given interval.

16. [latex]r={e}^{3\theta }\text{ on the interval }0\le \theta \le 2[/latex]
17. [latex]r=8+8\cos\theta \text{ on the interval }0\le \theta \le \pi[/latex]

For the following exercises (18-20), use the integration capabilities of a calculator to approximate the length of the curve.

18. [latex]r=3\theta \text{ on the interval }0\le \theta \le \dfrac{\pi }{2}[/latex]
19. [latex]r={\sin}^{2}\left(\dfrac{\theta }{2}\right)\text{ on the interval }0\le \theta \le \pi[/latex]
20. [latex]r=\sin\left(3\cos\theta \right)\text{ on the interval }0\le \theta \le \pi[/latex]

For the following exercise, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.

21. [latex]r=\sin\theta +\cos\theta \text{ on the interval }0\le \theta \le \pi[/latex]

For the following exercises (22-23), use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.

22. [latex]r=3\sin\theta \text{ on the interval }0\le \theta \le \pi[/latex]
23. [latex]r=6\sin\theta +8\cos\theta \text{ on the interval }0\le \theta \le \pi[/latex]

For the following exercises (24-29), find the slope of a tangent line to a polar curve [latex]r=f\left(\theta \right)[/latex]. Let [latex]x=r\cos\theta =f\left(\theta \right)\cos\theta[/latex] and [latex]y=r\sin\theta =f\left(\theta \right)\sin\theta[/latex], so the polar equation [latex]r=f\left(\theta \right)[/latex] is now written in parametric form.

24. Use the definition of the derivative [latex]\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{d}\theta }{\dfrac{dx}{d}\theta }[/latex] and the product rule to derive the derivative of a polar equation.
25. [latex]r=4\cos\theta[/latex]; [latex]\left(2,\dfrac{\pi }{3}\right)[/latex]
26. [latex]r=4+\sin\theta[/latex]; [latex]\left(3,\dfrac{3\pi }{2}\right)[/latex]
27. [latex]r=4\cos\left(2\theta \right)[/latex]; tips of the leaves
28. [latex]r=2\theta[/latex]; [latex]\left(\dfrac{\pi }{2},\dfrac{\pi }{4}\right)[/latex]
29. For the cardioid [latex]r=1+\sin\theta[/latex], find the slope of the tangent line when [latex]\theta =\dfrac{\pi }{3}[/latex].

For the following exercises (30-31), find the slope of the tangent line to the given polar curve at the point given by the value of [latex]\theta[/latex].

30. [latex]r=\theta[/latex], [latex]\theta =\dfrac{\pi }{2}[/latex]
31. Use technology: [latex]r=2+4\cos\theta[/latex] at [latex]\theta =\dfrac{\pi }{6}[/latex]

For the following exercises (32-33), find the points at which the following polar curves have a horizontal or vertical tangent line.

32. [latex]{r}^{2}=4\cos\left(2\theta \right)[/latex]
33. The cardioid [latex]r=1+\sin\theta[/latex]

Conic Sections

For the following exercises (1-4), determine the equation of the parabola using the information given.

1. Focus [latex]\left(4,0\right)[/latex] and directrix [latex]x=-4[/latex]
2. Focus [latex]\left(0,0.5\right)[/latex] and directrix [latex]y=-0.5[/latex]
3. Focus [latex]\left(0,2\right)[/latex] and directrix [latex]y=4[/latex]
4. Focus [latex]\left(-3,5\right)[/latex] and directrix [latex]y=1[/latex]

For the following exercises (5-8), determine the equation of the ellipse using the information given.

5. Endpoints of major axis at [latex]\left(4,0\right),\left(-4,0\right)[/latex] and foci located at [latex]\left(2,0\right),\left(-2,0\right)[/latex]
6. Endpoints of major axis at [latex]\left(0,2\right),\left(0,-2\right)[/latex] and foci located at [latex]\left(3,0\right),\left(-3,0\right)[/latex]
7. Endpoints of major axis at [latex]\left(-3,5\right),\left(-3,-3\right)[/latex] and foci located at [latex]\left(-3,3\right),\left(-3,-1\right)[/latex]
8. Foci located at [latex]\left(2,0\right),\left(-2,0\right)[/latex] and eccentricity of [latex]\dfrac{1}{2}[/latex]

For the following exercises (9-12), determine the equation of the hyperbola using the information given.

9. Vertices located at [latex]\left(5,0\right),\left(-5,0\right)[/latex] and foci located at [latex]\left(6,0\right),\left(-6,0\right)[/latex]
10. Endpoints of the conjugate axis located at [latex]\left(0,3\right),\left(0,-3\right)[/latex] and foci located [latex]\left(4,0\right),\left(-4,0\right)[/latex]
11. Vertices located at [latex]\left(-2,0\right),\left(-2,-4\right)[/latex] and focus located at [latex]\left(-2,-8\right)[/latex]
12. Foci located at [latex]\left(6,-0\right),\left(6,0\right)[/latex] and eccentricity of [latex]3[/latex]

For the following exercises (13-15), consider the following polar equations of conics. Determine the eccentricity and identify the conic.

13. [latex]r=\dfrac{-1}{1+\cos\theta }[/latex]
14. [latex]r=\dfrac{5}{2+\sin\theta }[/latex]
15. [latex]r=\dfrac{3}{2 - 6\sin\theta }[/latex]

For the following exercises (16-17), find a polar equation of the conic with focus at the origin and eccentricity and directrix as given.

16. [latex]\text{Directrix:}x=4;e=\dfrac{1}{5}[/latex]
17. [latex]\text{Directrix: y}=2;e=2[/latex]

For the following equations (18-20), determine which of the conic sections is described.

18. [latex]{x}^{2}+4xy - 2{y}^{2}-6=0[/latex]
19. [latex]{x}^{2}-xy+{y}^{2}-2=0[/latex]
20. [latex]52{x}^{2}-72xy+73{y}^{2}+40x+30y - 75=0[/latex]