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Using Polar Coordinates

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Using Polar Coordinates

Graphing and converting polar and rectangular coordinates

The grid at the left is a polar grid. The typical angles of 30o, 45o, 90o, … are shown on the graph along with circles of radius 1, 2, 3, 4, and 5 units.

A

Points in polar form are given as (r, ß ) where r is the radius to the point and ß is the angle of the point.

On one of your polar graphs, plot the point (3, 90o)?

The point on the graph labeled A is correct.

Now, try graphing .

C

A

Did you get point B?

Polar points have a new aspect. A radius can be negative! A negative radius means to go in the exact opposite direction of the angle.

B

To graph (-4, 240o), find 240o and move 4 units in the opposite direction. The opposite direction is always a 180o difference.

Point C is at (-4, 240o). This point could also be labeled as (4, 60o).

How would you write point A with a negative radius?

C

A

A correct answer would be (-3, 270o) or (-3, -90o).

In fact, there are an infinite number of ways to label a single polar point. Is (3, 450o) the same point?

B

Don’t forget, you can also use radian angles as well as angles in degrees.

On your own, find at least 4 different polar coordinates for point B.

On your own, find at least 4 different polar coordinates for point B.

C

A

There are many possible answers.Here are just a few.(2, 225o), (2, -135o)(-2, 45o), (-2, -315o)One could add or subtract 360o to any ofthe above angles.Radians would result in:(2, 5π/4), (2, -3π/4)(-2, π/4), (-2, -7π/4)One could add or subtract 2π to any previous radian angle.

B

Find the polar form for the rectangular point (4, 3).

To find the polar coordinate, we must calculate the radius and angle to the given point.

(4, 3)

r

We can use our knowledge of right triangle trigonometry to find the radius and angle.

3

ß

4

r2 = 32 + 42r2 = 25r = 5

tan ß = ¾ß = tan-1(¾)ß = 36.87o or 0.64 rad

The polar form of the rectangular point (4, 3) is (5, 36.87o)

In general, the rectangular point (x, y) is converted to polar form (r, θ) by:

1. Finding the radius

r2 = x2 + y2

(x, y)

2. Finding the angle

r

y

ß

tan ß = y/x or ß = tan-1(y/x)Recall that some angles require the angle to be converted to the appropriate quadrant.

x

Note: This is just like finding the length and direction angle of a vector!

On your own, find polar form for the point (-2, 3).

(-2, 3)

r2 = (-2)2 + 32r2 = 4 + 9r2 = 13r =

However, the angle must be in the second quadrant, so we add 180o to the answer and get an angle of 123.70o.

The polar form is ( , 123.70o)

Convert the polar point (4, 30o) to rectangular coordinates.

We are given the radius of 4 and angle of 30o. Find the values of x and y.

Using trig to find the values of x and y, we know that cos ß = x/r or x = r cos ß. Also, sin ß = y/r ory = r sin ß.

4

y

30o

x

The point in rectangular form is:

On your own, convert (3, 5π/3) to rectangular coordinates.

We are given the radius of 3 and angle of 5π/3 or 300o. Find the values of x and y.

-60o

The point in rectangular form is:

Equations in rectangular form use variables (x, y), whileequations in polar form use variables (r, ß) where ß is an angle.

Converting from one form to another involves changing the variables from one form to the other.

We have already used all of the conversions which are necessary.

Converting Polar to Rectangular

Converting Rectanglar to Polar

cos ß = x/rsin ß = y/rtan ß = y/xr2 = x2 + y2

x = r cos ßy = r sin ßx2 + y2 = r2

The goal is to change all x’s and y’s to r’s and ß’s.When possible, solve for r.

Example 1: Convert x2 + y2 = 16 to polar form.

Since x2 + y2 = r2, substitute into the equation.

r2 = 16

Simplify.

r = 4

r = 4 is the equivalent polar equation to x2 + y2 = 16

Example 2: Convert y = 3 to polar form.

Since y = r sin ß, substitute into the equation.

r sin ß = 3

Solve for r when possible.

r = 3 / sin ß

r = 3 csc ß is the equivalent polar equation.

Example 3: Convert (x - 3)2 + (y + 3)2 = 18 to polar form.

Square each binomial.

x2 – 6x + 9 + y2 + 6y + 9 = 18

Since x2 + y2 = r2, re-write and simplify by combining like terms.

x2 + y2 – 6x + 6y = 0

Substitute r2 for x2 + y2, r cos ß for x and r sin ß for y.

r2 – 6rcos ß + 6rsin ß = 0

Factor r as a common factor.

r(r – 6cos ß + 6sin ß) = 0

r = 0 or r – 6cos ß + 6sin ß = 0

Solve for r: r = 0 or r = 6cos ß – 6sin ß

The goal is to change all r’s and ß’s to x’s and y’s.

Example 1: Convert r = 4 to rectangular form.

Since r2 = x2 + y2, square both sides to get r2.

r2 = 16

Substitute.

x2 + y2 = 16

x2 + y2 = 16 is the equivalent polar equation to r = 4

Example 2: Convert r = 5 cos ß to rectangular form.

Since cos ß = x/r, substitute for cos ß.

Multiply both sides by r.

r2 = 5x

Substitute for r2.

x2 + y2 = 5x is rectangular form.

Example 3: Convert r = 3 csc ß to rectangular form.

Since csc ß = r/y, substitute for csc ß.

Multiply both sides by y/r.

Simplify

y = 3 is rectangular form.

6.4 / 1, 2, 7-13, 15, 16, 19, 20, 27, 28, 31-40