PRE-CALCULUS POLAR, VECTOR AND PARAMETRIC UNIT Polar Coordinates So far in mathematics, we have been working with what is called a, “____________________ Coordinate System.” Now, we will move into a “___________________ Coordinate System”. Coordinate: ( , ) ( , ) Graphing Polar Coordinates: With positive r: Plot ( 2 , 2 π 3 ) With negative r: Plot ( −3 , 5 π 4 ) Now, go backward and give 4 polar coordinates for the given point. Converting between Polar Coordinates and Rectangular Coordinates:
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PRE-CALCULUS POLAR, VECTOR AND PARAMETRIC UNITPolar Coordinates
So far in mathematics, we have been working with what is called a, “____________________ Coordinate System.” Now, we will move into a “___________________ Coordinate System”.
Coordinate: ( , ) ( , )
Graphing Polar Coordinates:
With positive r: Plot (2 , 2 π3 ) With negative r: Plot (−3 , 5π
4 )
Now, go backward and give 4 polar coordinates for the given point.
Converting between Polar Coordinates and Rectangular Coordinates:To convert, we must use:
cos (θ)= sin(θ)= tan(θ)=
So: x=¿ y=¿ x2+ y2=r2
Find the Cartesian coordinates of a points with polar coordinates
(4 , 5 π6 ) (−2 , π
2 )
Find the polar coordinates of the points:
(−3 ,−4). (0 ,3 )
Homework for Polar CoordinatesGraph the following polar coordinates:A . (1, π ) B .(−3 , π
3 )C .(2 , 5 π4 )D.(−3 , 11 π
6 )
Give 4 Polar Coordinates for each:
(, ) (, ) (, ) (, )
(, ) (, ) (, ) (, )
Convert to Rectangular Coordinates:
(−1 , 5 π3 ) (5 , 7 π
6 ) (−4 , π2 ) (6 , 3π
4 )
Convert to Polar Coordinates:
(−1 ,0 ) ( 12 , √32 ) (−2 ,5 )
PRE-CALCULUS POLAR, VECTOR AND PARAMETRIC UNIT
Graphing Polar Equations Two Basic Graphs:r=2 θ=π
4
Converting Rectangular Equations to Polar Equations:
3 Facts that must be used:x=r cos (θ ) y=x2+ y2=¿
The goal is to get an equation that looks like: r=¿ x2+ y2=6 y y=3 x+2
Converting from Polar Equations to Rectangular Equations:3 Facts that must be used:
sin (θ )= cos (θ )= r=√+¿¿
Goal is to convert everything to x’s and y’s:
r=2sin (θ) r= 31−2cos (θ)
Other Basic Graphs:r=4cos (θ) r=3sin(θ)
Homework for Graphing Polar EquationsConvert the given Cartesian equation to a polar equation:x=3 y=4 x2
x2+ y2=4 y x2− y2=x
Convert the given Polar Equation to a Cartesian Equation:r=3sin(θ) r=sec (θ)
r= 4sin (θ )+7 cos(θ)
Graph the following:r=3 θ=−2 π
3
r=−5 sin(θ) r=−3cos(θ)
PRE-CALCULUS POLAR, VECTOR AND PARAMETRIC UNITMore on Graphing Polar Equations
How to Graph a Polar Equation on a Graphics Calculator:1. Press the Mode button – go to scroll down to “polar” and press enter.2. Set Window Settings – for θ−min ,use 0 : for θ−min ,use2 π : for θ−scale , use .05
Problem Solving Procedure:1. Convert vectors to component form.2. Add the components of the vectors3. Convert back to length and direction if needed.
For directional problems, we use the map: Using the problems specs.
Going 3 miles north gives her a component of ⟨ , ⟩
From there, her magnitude is 2 with an angle of 315 °Converting to component form: x=2cos (315° ) , y=2sin(315 °)
Now, add the components together:Use Pythagorean Thm to find the length:
Use tan (θ )= yx to get the angle:
Homework for More with VectorsUsing the vectors given, compute u⃗+ v⃗ ,u⃗−v⃗∧2u⃗−3 v⃗
u⃗=⟨2 ,−3 ⟩, v⃗=⟨1,5 ⟩ u⃗=⟨−3,4 ⟩, v⃗=⟨−2,1 ⟩
A woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she and in what direction must she walk to head directly home?
In a scavenger hunt, directions are given to find a buried treasure. From a starting point at a flag pole, you must walk 30 ft. east, turn 30° to the north and travel 50 ft, and then turn due south and travel 75 ft. Find the components of each and calculate how far and in what direction you must go to get directly to the treasure from the flag pole.
v⃗1=⟨ , ⟩ v⃗2=⟨ , ⟩ v⃗3=⟨ , ⟩
v⃗∑ ¿= ⟨ , ⟩ ¿ Magnitude using Pyth Thm Angle:
An airplane is heading north at an airspeed of 600 km/hr, but there is a wind blowing from the southwest at 80 km/hr. How many degrees does the wind push the plane and what is the plane’s real speed because of this wind?
PRE-CALCULUS POLAR, VECTOR AND PARAMETRIC UNITParametric Equations
Another way of representing situations is to introduce a third _____________________ (or Parameter). This variable is the letter _____ which represents ______________________.
The advantages of using parametric equations:
Sketch a graph of:x (t )=t 2+1 y ( t )=2+t
Sketch a graph of: x (t )=2cos ( t ) y=3sin (t )¿0≤ t ≤2πInitial Point: __________ Terminal Point: __________
Converting from Parametric to Cartesian Equations:Using the equations we started with: x (t )=t 2+1 y (t )=2+t
Solve one equation for_____, then substitute into the other for ______.
Another way: x (t )=t 3 y ( t )=t 6
Converting using trigonometry:
Using the set of parametrics from earlier: x (t )=2cos ( t ) y=3 sin (t )We must use the Pythagorean identity: cos2(t)+sin2(t )=1
Isolate each trig function in the above set:
Substitute in to the Identity:
Homework for Parametric Equations
Sketch the parametric equations for the given domain of t.x (t )=1+2 t y ( t )=t 2;−2≤t ≤2 x (t )=4sin (t ) y ( t )=3cos ( t ); 0≤ t ≤2π
Write as a Cartesian: (Solve for t in the first equation)x (t )=2 t+1 y ( t )=3√ t x (t )=√t+2 y ( t )= log(t)
x (t )=2et y ( t )=1−5 t x (t )=e2 t y ( t )=e6 t
x (t )=5cos (t ) y (t )=6 sin(t) x (t )=1−cos (t ) y ( t )=√sin(t )
PRE-CALCULUS POLAR, VECTOR AND PARAMETRIC UNITMore with Parametric Equations
Converting from Cartesian to Parametric Equations (Parameterizing):Parameterize the following equations: y=3 x2+2x−1
x (t )= y (t )=¿
x= y3− y
y (t )=x (t )=¿
Constructing an xy graph based on the its parametric graphs.x(t) y(t)
The populations of rabbits and wolves on an island over time are given by the graphs below.
Use these graphs to sketch a graph in the r-w plane showing the relationship between the number of rabbits and the number of wolves.
A robot follows the path shown. Create a table of values for x(t) and y(t) functions. The robot takes one second to make each movement.
Homework for More with Parametrics
For each graph in the t-x and t-y plane, sketch a graph in the x-y plane.
Parameterize each Cartesian equation:y ( x )=3 x2+3 x ( y )=3 log ( y )+ y
x2
4+ y2
9=1 (Hint: think trig!) x2
49+ y2
81=1
Parameterize each graph. (Hint: first write the equation of the graph.)