START UP DAY 62 Kirby hits a ball when it is 4 ft above the ground with an initial velocity of 120 ft/sec. The ball leaves the bat at a 30° angle with the horizontal and heads toward a 30-ft fence 350 ft from home plate. (a) Does the ball clear the fence? (b) If so, by how much does it clear the fence? If not, could the ball be caught? Parameterize the line through (2, -3) and (5, 1)

http://youtu.be/rHnbyuBSLRQ http://youtu.be/Qx1CnIXesMk OBJECTIVE: SWBAT convert from a polar equation to a rectangular equation with and without the use of technology. ESSENTIAL QUESTIONS: What are polar coordinates? What is the Polar Axis? Why are there multiple representations for the same point? How do we convert polar coordinates to rectangular? How do we convert rectangular to polar? HOME LEARNING:p.492‐493; 1, 4, 5, 7, 10,19, 22, 23, 26, 31-34, 35, 38, 42, 43, 48 and 51 http://youtu.be/rHnbyuBSLRQ http://youtu.be/Qx1CnIXesMk

Definition of Polar Coordinates Polar Coordinates are two values that locate a point on a plane by its distance from a fixed pole and its angle from a fixed line passing through the pole

The Polar Plane POLE POLAR AXIS

EXAMPLE: GET the POINT? Given: (2, л/3)

Get the Point? (2, л/3) Move out to the circle of radius “r” along the POLAR AXIS ( or the x-axis) Use “θ”, in STANDARD POSTION, to rotate to get to the point.

Try a few and see what you can do! P (3, – /6) O (-2, 3 /2) L (3, - 45°) A (-1, 5 /3) R (4, 150°)

Multiple Representations for the same point? The coordinates (r,  ) and (r,  + 2n) represent the same point. Another way to obtain multiple representations of a point is to use negative values for r. Because r is a directed distance, the coordinates (r,  ) and (–r,  + ) represent the same point

Polar Rectangular POLAR RECTANGULAR (0,0) is called the origin Coordinates are in form (x,y) (0,0) is called the pole Coordinates are in form (r, θ)

POLAR Rectangular The hypotenuse is the radius with a length of r. The sides are x (adjacent) and y (opposite). By using these properties, we get that: x = r cos θ & y = r sin θ where r2= x2+y2 and tan θ = y/x 2, л/3 r y x

How can we go back? Rectangular Polar

Rectangular Polar FIRST determine your “circle” by finding “r” NEXT, determine your angle of rotation, or “θ” ( Just remember that it will be easier to see as a degree and you may have to adjust for quadrant location)

Think it through... What should you do? Convert each of the following:

Check your work….

An Exploration In Polar Mode    

POLAR Exploration—Do you see what I see?

Lines & Circles….The easy Polar Equations Consider the following polar equations. Create a table of values and sketch the graphs: r θ -2 π/4 2 5 r θ 3

Algebraically? If you have a “θ=“… just take the tangent of both sides. Replace tan θ with y/x Evaluate the tangent of your specific angle. Isolate y

Algebraically? If you have a “r=“… just square both sides. Replace r squared with.. Now you’ve got your equation in standard form.

POLAR TO RECTANGULAR CONVERT each of following Polar Equations: r = 4 sin θ r = sec θ tan θ Graphically? Algebraically?

Simplify and write in Standard Form POLAR TO RECTANGULAR Cross multiply Algebraically Distribute Substitute for “x” Isolate “r” Square both sides Expand Substitute for “r2” Simplify and write in Standard Form

More Equation Conversions To convert a rectangular equation to polar form, you simply replace x with r cos  and y with r sin . For instance, the rectangular equation y = x2 can be written in polar form as follows. y = x2 r sin  = (r cos  )2 Rectangular equation Polar equation Isolate “r” and Simplify using basic trig identities sec  tan  = r

Try These Polar Equations to Rectangular Form Describe the graph of each polar equation and find the corresponding rectangular equation. a. r = 2 b. c. r = sec 

POLAR RECTANGULAR CONVERT TO RECTANGULAR—ALGEBRAICALLY! CONVERT TO POLAR—ALGEBRAICALLY!

POLAR RECTANGULAR

POLAR RECTANGULAR

Let’s Try it! State 2 other pairs of polar coordinates for the point (3, 2π/3) Convert the rectangular point (-3, -3) to polar coordinates. Complete each basic identity: https://youtu.be/sKvcsymsRrU