Using Polar Coordinates Graphing and converting polar and rectangular coordinates

Graphing Polar Coordinates The grid at the left is a polar grid. The typical angles of 30 o, 45 o, 90 o, … are shown on the graph along with circles of radius 1, 2, 3, 4, and 5 units. 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, 90 o )? A The point on the graph labeled A is correct.

Graphing Polar Coordinates Now, try graphing. 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. A To graph (-4, 240 o ), find 240 o and move 4 units in the opposite direction. The opposite direction is always a 180 o difference. B C Point C is at (-4, 240 o ). This point could also be labeled as (4, 60 o ).

Graphing Polar Coordinates How would you write point A with a negative radius? A correct answer would be (-3, 270 o ) or (-3, -90 o ). In fact, there are an infinite number of ways to label a single polar point. Is (3, 450 o ) the same point? A Don’t forget, you can also use radian angles as well as angles in degrees. B C On your own, find at least 4 different polar coordinates for point B.

Graphing Polar Coordinates On your own, find at least 4 different polar coordinates for point B. There are many possible answers. Here are just a few. (2, 225 o ), (2, -135 o ) (-2, 45 o ), (-2, -315 o ) One could add or subtract 360 o to any of the 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. A B C

Converting from Rectangular to Polar 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) We can use our knowledge of right triangle trigonometry to find the radius and angle. ß 3 4 r r 2 = r 2 = 25 r = 5 tan ß = ¾ ß = tan -1 (¾) ß = o or 0.64 rad The polar form of the rectangular point (4, 3) is (5, o )

Converting from Rectangular to Polar In general, the rectangular point (x, y) is converted to polar form (r, θ) by: 1. Finding the radius (x, y) r 2 = x 2 + y 2 ß y x r 2. Finding the angle tan ß = y/x or ß = tan -1 (y/x) Recall that some angles require the angle to be converted to the appropriate quadrant. Note: This is just like finding the length and direction angle of a vector!

However, the angle must be in the second quadrant, so we add 180 o to the answer and get an angle of o. The polar form is (, o ) r 2 = (-2) r 2 = r 2 = 13 r = Converting from Rectangular to Polar On your own, find polar form for the point (-2, 3). (-2, 3)

Converting from Polar to Rectanglar Convert the polar point (4, 30 o ) to rectangular coordinates o We are given the radius of 4 and angle of 30 o. 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 or y = r sin ß. x y The point in rectangular form is:

Converting from Polar to Rectanglar On your own, convert (3, 5π/3) to rectangular coordinates. -60 o We are given the radius of 3 and angle of 5π/3 or 300 o. Find the values of x and y. The point in rectangular form is:

Rectangular and Polar Equations Equations in rectangular form use variables (x, y), while equations 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 cos ß = x/r sin ß = y/r tan ß = y/x r 2 = x 2 + y 2 Converting Rectanglar to Polar x = r cos ß y = r sin ß x 2 + y 2 = r 2

Convert Rectangular Equations to Polar Equations 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 x 2 + y 2 = 16 to polar form. Since x 2 + y 2 = r 2, substitute into the equation. r 2 = 16 Simplify. r = 4 r = 4 is the equivalent polar equation to x 2 + y 2 = 16

Convert Rectangular Equations to Polar Equations 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.

Convert Rectangular Equations to Polar Equations Example 3: Convert (x - 3) 2 + (y + 3) 2 = 18 to polar form. Square each binomial. x 2 – 6x y 2 + 6y + 9 = 18 Since x 2 + y 2 = r 2, re-write and simplify by combining like terms. x 2 + y 2 – 6x + 6y = 0 Substitute r 2 for x 2 + y 2, r cos ß for x and r sin ß for y. r 2 – 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

Convert Polar Equations to Rectangular Equations 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 r 2 = x 2 + y 2, square both sides to get r 2. r 2 = 16 Substitute. x 2 + y 2 = 16 x 2 + y 2 = 16 is the equivalent polar equation to r = 4

Convert Polar Equations to Rectangular Equations Example 2: Convert r = 5 cos ß to rectangular form. Since cos ß = x/r, substitute for cos ß. Multiply both sides by r. r 2 = 5x Substitute for r 2. x 2 + y 2 = 5x is rectangular form.

Convert Polar Equations to Rectangular Equations 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.

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