MAT 224Name

Johns – SCCC

Polar Coordinates – Notes

General Idea

Conversion Formulas

Cartesian to Polar:Polar to Cartesian:

r2 = x2 + y2x = r cos 

tan  = y = r sin

Nuts and Bolts of Cartesian and Polar Coordinate Systems

(x,y) = (2,3) is the intersection(r,) = (4, ) is the intersection

of the line x = 2 and the line y = 3of the circle r = 4 and the ray

 = .

Equations of Common Geometric Features

Unit Vectors in Cartesian and Polar Coordinates

is a unit vector orthogonal to the line x = 2is a unit vector orthogonal to the

pointing in the direction of increasing x.circle r = 4 pointing in the direction of

increasing r.

is a unit vector orthogonal to the line y = 3 is a unit vector orthogonal to the ray

pointing in the direction of increasing y. = in the direction of increasing.

MAT 224Name

Johns – SCCC

Polar Coordinates Worksheet

Do all of the following problems on cartesian graph paper.

1. On the same set of cartesian axes, draw and label (with their letter) the following points:

A: (x, y) = (0, -2)B: (x, y) = (-2, -2) C: (x, y) = (4,-3) D: (x, y) = (-3,2)

Then draw the polar unit vectors and at each of these points.

2. Find the equivalent polar coordinates (r, ) for each of the points inQuestion #1. First, give exact

answers. Then check your exact answers by finding approximate answers to the nearest tenth.

3. By inspection, state the polar unit vectors and in terms of cartesian unit

vectors and at each of the points in Question #1. Give exact answers.

4. On a different set of cartesian axes, draw the following pointsgiven in polar coordinates.

(r, ) = (4, ) (r,) = (3, 4)(r, ) = (2, 2)

Then write the exact cartesian (x, y) coordinates next to each point.

5. Find a polar equation for the following features:

a) a circle centered at the origin passing through point (6,8)

b) the negative y-axis

c) the line y = 3

d) the line y = 2x

6. a) Show that the polar equation of a circle with center at (0,1) and radius of 1 is r = 2 sin .

b) On a different cartesian set of axes, draw a fairly large sketch of this circle.

c) By choosing different  values, we can use the equation in part a) to find coordinates (r, ) and plot

them as points on our circle. For the following  values, find the accompanying r values and label

these points on the circle with the given letter below.

E:  = 0F:  = G:  = H:  = K:  = 

d) Something strange happens when we put values between  and 2 in our equation in part a): we get

negative r values! For example, if  = , then r = −1. Since this (r, ) pair must still lie on the

circle, explainin a sentence what a r value of −1 means. Which of the points in part c) is at the same

locationas (r, ) = (-1,)?

e) There is nothing stopping us from choosing a negative  value for the polar equation in part a).

If we choose  = , then r = -2. Which of the points in part c) is at the same location as

(r, ) = (-2,)?

7. Statefour different (r, ) pairs for the point (x, y) = (-1,1). One pair must have a negative r-value.

8. a) Therefore, most (x, y) locations have a countable infinity of (r,) pairs associated with them. But the

origin causes usmore problems. Explain why.

b) Does the origin have polar unit vectors? Explain.

9. Consider the two polar unit vectors and drawn at the point (r, ) below.

a) Carefully explain where the formula

= cos + sin comes from. Hint: the diagram may help.

b) Now carefully explain where the formula

= −sin + cos comes from. There are many possible explanations – just pick one and explain well.

c) Verify these formulas by applying them to the points in Question #1 and comparing the results with your answers for Question #3.

10. a) An advantage of the cartesian coordinate system is that its unit vector does not change direction as x or y increases (ie. and ) . Explain geometrically why this is so. (HINT: recall that at point (x, y) = (a, b), its unit vector is orthogonal to line x = a. What happens to as x increases? as y increases?) NOTE: unit vector also does not change direction as x or y increases!

b) Now consider the polar unit vector at point (r, ) = (c, d). Since is orthogonal to the circle r = c and pointing away from the origin, if radius r increases, unit vector will still point away from the origin in the same direction but will be orthogonal to a bigger circle. Therefore, as r increases, unit vector does not change (ie. ). Yet if angle  increases(assuming r doesn’t change), the unit vector rotates slightly counterclockwise so that is a non-zero vector in the direction of unit vector . Draw rough sketchesillustrating what happens to unit vector when i) r increases and ii) when  increases .

c) Using the formulas in Question #9 and taking derivatives, show that and that ,

a unit vector in the increasing  direction.

d) Now consider the polar unit vector at point (r, ) = (c, d). Recall that is tangent to the circle r = c and pointing in the direction of increasing .

i) First, draw a picture of what happens to when r increases. What is ? Explain

geometrically.

ii) Next, draw a picture of what happens to when  increases. In which direction does the vector

point?

e) Using the formulas in Question #9 and taking derivatives, show that and that ,

a unit vector in the direction of decreasing r.

[What does this mean? In the cartesian coordinate system, the unit vectors and remain unchanged as x or y increases; so we don’t have to differentiate them wrt x or y when taking derivatives of vector functions given in cartesian coordinates.

In polar coordinates, the unit vectors and remain unchanged as r changes, but both unit vectors have a rate of change when  changes. Therefore, we do have to differentiate these unit vectors (at least wrt ) when taking derivatives of vector functions given in polar coordinates.]