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

Cardiods: The form: r=a+a sinθ

Sketch: r=2+2cos (θ)

Limacons: The form: r=a+b sin (θ )∨a+a cos(θ)

r=2+4sin (θ) r=2−3sin (θ ) r=2+24cos (θ) r=5−6cos (θ)

What relationships/characteristics can you deduce?

Roses: The form: r=sin a (θ )∨r=cos a(θ)

Sketch: r=sin 3θ r=sin 4θ r=cos6θ r=cos5θ

What relationships/characteristics can you deduce?

Homework for More on Graphing Polar EquationsMatch each equation with the graph without using a calculator.

Now, using a calculator, match the equation with the graph.

Using a graphics calculator, graph the following:

PRE-CALCULUS POLAR, VECTOR AND PARAMETRIC UNITComplex Numbers

The definition of a Imaginary Number:

i=i2=i3=i4=¿

Simplify:

√−49 √−72 i43 6 i18

The Definition of an Complex Number:A complex number takes on the form: z=¿

Where a is a ____________ number and bi is a _______________ number.

A Complex Plane:

Plot 3 + 4i on a complex plane.

Operations with Complex Numbers:

(3−4 i )−(1−2 i) 4 i(−3+7 i) −4 i(5i)

(3−2i)(4+3 i) (2−7i)(2+7 i) 1−6 i5 i

4−3 i2+5 i

Homework for Complex NumbersSimplify each expression to a single complex number.

√−121 √−98 √−6√−12 2+√−122

Graph the following on the complex plane below:

A. −3 i B. 3−4 i C. −2 D. −3+2 i

Simplify the following into a single complex number:(3+2 i )+(5−3 i) (−5+3 i )−(6−i)

(2+3 i)(4 i ) (6−2 i)(5)

(2−3i)(4−i) (4−2i)(4+2i )

3+4 i2 −5+3 i

2 i

3+2i4−i 1−i

1+i

PRE-CALCULUS POLAR, VECTOR AND PARAMETRIC UNITPolar Forms of Complex Numbers

Recall the facts behind Polar Coordinates:cos (θ)= sin(θ)= tan(θ)=

So: x=¿ y=¿ x2+ y2=r2

Also, recall that z=¿

Now substituting in for x and y: z=¿

In the 18th century, Leonhard Euler demonstrated a relationship that allowed complex numbers to greatly simplify trig calculations:

The Polar Form of a Complex Number: z=r e iθ

Euler’s Formula: r e iθ=r cos (θ )+i r sin(θ)

Find the Polar Form of the following Complex Numbers: 4 i −3 −4+4 i √3+i

Sketch, then find the value for r and for θ:

4 i=e −3=e −4+4 i=e √3+i

Writing Polar Form back into Complex Numbers:

3eπ6 i=cos( )+i sin( )

Evaluate: (−4+4 i )4 (from the earlier example: −4+4 i=e) Now sub in:Homework for Polar Forms of Complex Numbers

Rewrite each complex number from polar form into a + bi form.

3eπ4 i 4 e

π3 i

8e5π4 i

Rewrite each complex number into polar form r e iθ form. 5 -3i 2 + 2i -3 – 3i

5 + 3i -1 – 4i

Compute each of the following, simplifying the result into a + bi form.(2+2 i )8 √−3+3 i

PRE-CALCULUS POLAR, VECTOR AND PARAMETRIC UNITVectors

A vector has both _________________ and ________________________.A vector is represented by an arrow and looks like a ________ from geometry.

A⃗Brepresents a vector with starting point ____, moving towards ____Other ways to represent a vector: u⃗∨v⃗

Drawing a Vector: Find a vector that represents the movement from point P(-1,2) to point Q(3,3).

Adding and Subtracting Vectors Geometrically:The sum u⃗+ v⃗ , draw v⃗ starting from the end of u⃗ .

Given the two vectors shown below, draw u⃗+ v⃗ .

Scaling Vectors: To scale a vector by a constant, say: 3 u⃗. It will be a vector

To scale a vector, − v⃗ . It will be a vector.

Given the vectors shown, draw u⃗−v⃗.

Given the vector shown, draw 3 u⃗ , and −2 u⃗

Component Form of Vectors:It is most useful to represent a vector using length and an angle θ , usually measured

from ____________________ position.

A vector of length 1 is called a _________ vector.

Component Form of Representing a Vector:The vector u⃗=⟨ x , y ⟩ is a vector with starting point at (0,0) and ending point at (x,y).

Draw the vector u⃗=⟨3,4 ⟩

Find the Component Form of a vector with length 5 at an angle of 135°.Sometimes its better to use the polar conversions x=r cos (θ )∧ y=¿

Component Form: ⟨ , ⟩Find the magnitude and angle θ representing the vector u⃗=⟨3 ,−2 ⟩

Homework for VectorsWrite the given vectors in Component Form.

Given the vectors shown, sketch: u⃗+ v⃗ ,u⃗−v⃗

From the given magnitude and direction in standard position, write the vector in component form.

Magnitude of 6, Direction of 45 °. Magnitude of 8, Direction of 220 °

From the given component form, find the magnitude and direction of the vector.⟨0,4 ⟩ ⟨6,6 ⟩ ⟨−2,1 ⟩ ⟨2,−5 ⟩

PRE-CALCULUS POLAR, VECTOR AND PARAMETRIC UNITMore with Vectors

Adding and Scaling Vectors in Component Form

Given: u⃗=⟨3 ,−2 ⟩∧ v⃗= ⟨−1,4 ⟩ , find anewvector w⃗=3 u⃗−2 v⃗

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.)