Precalculus Polar Coordinates and Parametric Equations Worksheet – PDF

Precalculus Polar Parametric Worksheet: problems 1–5

1. Read points on a polar grid

Read the polar coordinates \((r,\theta)\) of points A, B, C and D from the graph. Use \(r> 0\) and \(0^{\circ}\le\theta< 360^{\circ}\).

2. Convert polar to rectangular

Convert each polar point to rectangular coordinates.

1. \((4,\frac{\pi}{3})\)
2. \((6,\frac{3\pi}{4})\)
3. \((5,\pi)\)
4. \((2,-\frac{\pi}{2})\)

3. Convert rectangular to polar

Convert each rectangular point to polar coordinates with \(r> 0\) and \(0\le\theta< 2\pi\).

1. \((3,3\sqrt{3})\)
2. \((-4,0)\)
3. \((-5,5)\)
4. \((2,-2\sqrt{3})\)

4. Rewrite equivalent polar pairs

Rewrite each polar pair so that \(r> 0\) and \(\theta\) is between 0 and one full turn, keeping the same units.

1. \((-2,70^{\circ})\)
2. \((5,-45^{\circ})\)
3. \((-4,\frac{\pi}{6})\)
4. \((3,5\pi)\)

5. Distance between polar points

Find the exact distance between the two points. Use \(d^{2}=r_1^{2}+r_2^{2}-2r_1r_2\cos(\theta_2-\theta_1)\).

1. \((4,20^{\circ})\) and \((6,80^{\circ})\)
2. \((5,10^{\circ})\) and \((5,130^{\circ})\)

Precalculus Polar Parametric Worksheet: problems 6–10

6. Rectangular equations to polar form

Write each rectangular equation in polar form.

1. \(x=4\)
2. \(y=x\)
3. \(x^{2}+y^{2}=25\)
4. \(x^{2}+y^{2}=6y\)

7. Polar equations to rectangular form

Write each polar equation as a rectangular equation.

1. \(r=7\)
2. \(\theta=\frac{\pi}{3}\)
3. \(r=8\sin\theta\)
4. \(r=2\cos\theta+2\sin\theta\)

8. Identify a polar circle

State the center and radius of the circle graphed for \(r=10\cos\theta\), then write its rectangular equation.

9. Complete a cardioid table

Complete the table for \(r=2+2\cos\theta\). Then name the graph and state the greatest value of \(r\).

\(\theta\)	0	\(\frac{\pi}{3}\)	\(\frac{\pi}{2}\)	\(\frac{2\pi}{3}\)	\(\pi\)	\(\frac{4\pi}{3}\)	\(\frac{3\pi}{2}\)	\(\frac{5\pi}{3}\)
\(r\)

10. Classify polar graphs

Name the type of graph for each equation. For a limaçon, say whether it has an inner loop, a dimple or neither. For a rose, give the number of petals.

1. \(r=3+3\sin\theta\)
2. \(r=2+4\cos\theta\)
3. \(r=3+2\cos\theta\)
4. \(r=5+2\cos\theta\)
5. \(r=5\cos 3\theta\)
6. \(r=3\sin 2\theta\)

Precalculus Polar Parametric Worksheet: problems 11–15

11. Find the tips of a rose

Use the graph of \(r=5\cos 3\theta\) to state the number of petals and the petal length. Then find every \(\theta\) with \(0\le\theta< 2\pi\) for which \(r=5\).

12. Pole crossings of a rose

For \(r=4\cos 2\theta\), find every \(\theta\) in \([0,2\pi)\) at which the graph passes through the pole. Then state the number of petals and the angles where \(r=4\).

13. Limaçon with an inner loop

For \(r=2+4\cos\theta\), find the greatest value of \(r\), the angles in \([0,2\pi)\) at which the graph passes through the pole, and the rectangular point that corresponds to \(\theta=\pi\).

14. Make a table of parametric points

For \(x=2t+1\) and \(y=t^{2}-3\), find the point for each value \(t=-2,-1,0,1,2\). Plot the points on the grid and connect them in order of increasing \(t\).

15. Eliminate the parameter, linear

Eliminate the parameter and write \(y\) in terms of \(x\).

1. \(x=3t-2\), \(y=t+4\)
2. \(x=5-2t\), \(y=4t+1\)

Precalculus Polar Parametric Worksheet: problems 16–20

16. Eliminate the parameter, quadratic

Eliminate the parameter and identify the curve. Let \(t\) be any real number.

1. \(x=t^{2}\), \(y=2t-1\)
2. \(x=t+3\), \(y=t^{2}-6t+5\)

17. Eliminate the parameter, ellipse

For \(x=4\cos t\) and \(y=3\sin t\) with \(0\le t\le 2\pi\), eliminate the parameter, then find the points for \(t=0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\).

18. Shifted circle from parametric equations

Eliminate the parameter for a point that moves along \(x=2+3\cos t\), \(y=-1+3\sin t\), \(0\le t\le 2\pi\). Then state the center and radius, and find the starting point and the point at \(t=\pi\).

19. Soccer kick

Answer each part for a soccer ball kicked by Maya. Its position after \(t\) seconds is \(x=24t\), \(y=40t-16t^{2}\), in feet.

1. Find the height of the ball at \(t=1\).
2. Find the time at which the ball lands.
3. Find how far the ball travels horizontally before it lands.
4. Find the maximum height of the ball.
5. Eliminate the parameter to write \(y\) in terms of \(x\).

20. Ferris wheel

Answer each part about the ride Diego takes on a Ferris wheel of radius 30 ft whose center is 35 ft above the ground. His position is \(x=30\cos\frac{\pi t}{10}\), \(y=35+30\sin\frac{\pi t}{10}\), with \(t\) in seconds.

1. Find the time for one full turn.
2. Find his position at \(t=5\).
3. Find his lowest height above the ground.
4. Eliminate the parameter.
5. Find the first time his height is 50 ft.

Precalculus Polar Parametric Worksheet: problems 21–22

21. Radar stations

A radar station at the pole detects two boats. Boat 1 is at \((12\text{ mi},40^{\circ})\) and boat 2 is at \((9\text{ mi},100^{\circ})\). Find the rectangular coordinates of boat 1 to the nearest hundredth, and find the exact and approximate distance between the boats.

22. Rose flower bed

Answer each part for the flower bed Priya designs, shaped like the rose \(r=8\sin 3\theta\), with distances in feet and the pole at the center.

1. How many petals does the bed have?
2. Find the angles in \([0,2\pi)\) at which the petal tips lie.
3. Find the distance between two neighboring tips.
4. Priya runs edging along the straight segments joining the three tips. Find the total length of edging.

What should you do first on this worksheet?

Begin with Problems 1 to 5, because they build the basic skill. In Problem 2, convert \((4,\frac{\pi}{3})\) with \(x=4\cos\frac{\pi}{3}\). In Problem 5, the distance formula uses the angle gap, so subtract the angles first. Then move to the equation problems.

How do you graph polar equations like roses and limaçons?

Problem 9 asks for a table of \(r=2+2\cos\theta\). The greatest value is 4 at \(\theta=0\), and the curve is a cardioid. Problem 11 uses \(r=5\cos 3\theta\), so expect 3 petals of length 5. Problem 13 uses \(r=2+4\cos\theta\); because \(2<4\), an inner loop appears.

What mistakes appear in polar problems?

• Writing \(\theta\) in degrees while the calculator uses radians.
• Ignoring the quadrant in Problem 3, which gives the wrong angle.
• Forgetting that \(x^2+y^2=6y\) in Problem 6 becomes \(r=6\sin\theta\) after dividing by \(r\).

Check each answer by converting back.

How do parametric equations model real motion?

Problem 19 gives a kick with \(y=40t-16t^2\). Set \(y=0\) to find the landing time, then use \(x=24t\) for the distance. Problem 20 describes a wheel of radius 30 ft, so the height uses a cosine or sine term plus 35.

How can you plot a rose from a table?

For Problem 12, list \(\theta\) in steps of \(\frac{\pi}{8}\) and compute \(r=4\cos 2\theta\). Plot each pair, then connect the points smoothly. Positive values draw the petals along the axes, while negative values swing to the opposite side.

How should you review your results?

Mark each missed problem by topic: points, conversions, graphs or parameters. Redo only that group the next day, because short repeats work better than a long marathon. Afterward, compare with the answer key and read the explanation for each error.

See every chapter on the Precalculus math page.