6/6/13 Obj: SWBAT review for Semester Final Bell Ringer: Turn in Polar Worksheets HW: Review Packet #16-30 Announcements: Take missing Quizzes Last day.

6/6/13 Obj: SWBAT review for Semester Final

Bell Ringer:

Turn in Polar Worksheets

HW: Review Packet #16-30

Announcements:• Take missing Quizzes• Last day to bring books back is day of the Final• Turn in 3 missing assignments partial credit; by Friday• Check grades to look for missing assignments• Final Exam Tuesday Bring in books on day of exam

Quick Complete the Square Example

(r, )

You are familiar with plotting with a rectangular coordinate system.

We are going to look at a new coordinate system called the polar coordinate system.

The center of the graph is called the pole.

Angles are measured from the positive x axis.

Points are represented by a radius and an angle

(r, )radius angle

To plot the point

4,5

First find the angle

Then move out along the terminal side 5

A negative angle would be measured clockwise like usual.

To plot a point with a negative radius, find the terminal side of the angle but then measure from the pole in the negative direction of the terminal side.

4

3,3

3

2,4

Let's plot the following points:

2,7

2,7

2

5,7

2

3,7

Notice unlike in the rectangular coordinate system, there are many ways to list the same point.

Let's take a point in the rectangular coordinate system and convert it to the polar coordinate system.

(3, 4)

r

Based on the trig you know can you see how to find r and ?

4

3r = 5

222 43 r

3

4tan

93.03

4tan 1

We'll find in radians

(5, 0.93)polar coordinates are:

Let's generalize this to find formulas for converting from rectangular to polar coordinates.

(x, y)

r y

x

222 ryx

x

ytan

22 yxr

x

y1tanYou need to consider the quadrant in

which P lies in order to find the value of .

Now let's go the other way, from polar to rectangular coordinates.

Based on the trig you know can you see how to find x and y?

44cos

x

rectangular coordinates are:

4,4

4 yx4

222

24

x

44sin

y

222

24

y

2

2,

2

2

Let's generalize the conversion from polar to rectangular coordinates.

r

xcos

,rr yx

r

ysin

cosrx

sinry

330315

300

270240

225

210

180

150

135

120

0

9060

30

45

Polar coordinates can also be given with the angle in degrees.

(8, 210°)

(6, -120°)

(-5, 300°)

(-3, 540°)

12

To find the rectangular coordinates for a point given its polar coordinates, we can use the trig functions.

4,3

Example

13

14

Likewise, we can find the polar coordinates if we are given the rectangular coordinates using the trig functions. Think about the Pythagorean Theorem.

Example:

Find the polar coordinates (r, θ) for the point for which the rectangular coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

15

Conversion from Rectangular Coordinates to Polar Coordinates

If P is a point with rectangular coordinates (x, y), the polar coordinates

(r, ) of P are given by

2 2 1tanref

yr x y

x

P

You need to consider the quadrant in

which P lies in order to find the value of .

16

) 2, 2a

Find polar coordinates of a point whose rectangular coordinates are given. Give exact answers with θ in degrees.

17

) 1, 3b

Find polar coordinates of a point whose rectangular coordinates are given. Give exact answers with θ in degrees.

18

The TI-84 calculator has handy conversion features built-in. Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)

6: Returns value of given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )

8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).

922 yx

Convert the rectangular coordinate system equation to a polar coordinate system equation.

22 yxr 3r

r must be 3 but there is no restriction on so consider all values.

Here each r unit is 1/2 and we went out 3

and did all angles.

? and torelated was

how s,conversion From22 yxr

Before we do the conversion let's look at the graph.

Convert the rectangular coordinate system equation to a polar coordinate system equation.

yx 42

cosrx

sinry sin4cos 2 rr

sin4cos22 rr

substitute in for x and y

We wouldn't recognize what this equation looked like in polar coordinates but looking at the rectangular equation we'd know it was a parabola.

What are the polar conversions we found for x and y?

21

Write Polar Equation in Rectangular Form

Given r = 2 sin θ– Write as rectangular

equation

Use definitions– And identities

– Graph the given equation for clues

1

2 2 2

cos

sin

tan

x r

y r

y

x

r x y

22

Write Polar Equation in Rectangular Form

Given r = 2 sin θ

– We know

– Thus

– And

2

2 2

sin2

2

2

r y

r

r y

x y y

23

Write Rectangular Equation in Polar Form

Consider 2x – 3y = 6

– As before, usedefinitions

1

2 2 2

cos

sin

tan

x r

y r

y

x

r x y

2 cos 3 sin 6

2cos 3sin 6

6

2cos 3sin

r r

r

r

24

Polar Coordinates

How do we graph or use a graph to describe loops and curves?

Background The use of polar coordinates allows for the analysis of families of curves difficult to handle through rectangular coordinates (x,y). If a curve is a rectangular coordinate graph of a function, it cannot have any loops since, for a given value there can be at most one corresponding value. However, using polar coordinates, curves with loops can appear as graphs of functions.

http://www.xpmath.com/careers/topicsresult.php?subjectID=4&topicID=18

25

26

Polar Coordinates (Sect 21.10)

A point P in the polar coordinate system is represented by an ordered pair .

• If , then r is the distance of the point from the pole.

• is an angle (in degrees or radians) formed by the polar axis and a ray from the pole through the point.

,r

0r

27

Example

Plot the point P with polar coordinates 2, .4

28

Polar Coordinates

If , then the point is located units on the ray that extends in the opposite direction of the terminal side of .

For , r = |r| and ϴ = ϴ + 180

Ex. (-7, 70⁰) = (7, 250)

0r r

0r

29

Example

Plot the point with polar coordinates

4,3

30

5) 3,

3a

) 2,4

b

Plotting Points Using Polar Coordinates

31

) 3,0c ) 5,2

d

Plotting Points Using Polar Coordinates

32

A) B)

C) D)

33

• Using polar coordinates, the same point can be described by many different representations. Which of the following do(does) not describe the point (8,60°) ?

(8,−60°) (8,-300°) (8,420°) (-8,-120°)

• Using polar coordinates, write 3 more

representations of the point (5,150°)

34

Graphing in polar coordinates• Hit the MODE key. • Arrow down to where it says Func

(short for "function" which is a bit misleading since they are all functions). • Now, use the right arrow to choose Pol. • Hit ENTER.

(*It's easy to forget this step, but it's crucial: until you hit ENTER you have not actually selected Pol, even though it looks like you have!)

The calculator is now in polar coordinates mode. To see what that means, try this. Hit the Y= key. Note that, instead of Y1=, Y2=, and so on, you now have r1= and so on. In the r1= slot, type 5-5sin( Now hit the familiar X,T,q,n key, and you get an unfamiliar result. In polar coordinates mode, this

key gives you a ϴ instead of an X. Finally, close off the parentheses and hit GRAPH.

If you did everything right, you just asked the calculator to graph the polar equation r=5-5sin(ϴ). The result looks a bit like a valentine.

The WINDOW options are a little different in this mode too. You can still specify X and Y ranges, which define the viewing screen. But you can also specify the ϴvalues that the calculator begins and ends with; for instance, you may limit the graph to 0< ϴ <π/2. This would not change the viewing window, but it would only draw part of the graph.

35

End here 6/3

36

End of Section

37

Definitions of Trigonometric Functions of Any Angle

Let be an angle in standard position with (x, y) a point on the terminal side of and

Definitions of Trig Functions of Any Angle(Sect 8.1)

2 2r x y

sin csc

cos sec

tan cot

y r

r y

x r

r xy x

x y

y

x

(x, y)

r

38

Since the radius is always positive (r > 0), the signs

of the trig functions are dependent upon the signs of

x and y. Therefore, we can determine the sign of

the functions by knowing the quadrant in which the

terminal side of the angle lies.

The Signs of the Trig Functions

39

The Signs of the Trig Functions

40

Where each trig function is POSITIVE:

A

CT

S

“All Students Take Calculus”

Translation:

A = All 3 functions are positive in Quad 1

S= Sine function is positive in Quad 2

T= Tangent function is positive in Quad 3

C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is positive, secant is positive wherever cosine is positive, and cotangent is positive wherever tangent is positive.

41

Determine if the following functions are positive or negative:

Example

sin 210°

cos 320°

cot (-135°)

csc 500°

tan 315°

42

Examples

For the given values, determine the quadrant(s) in which the terminal side of θ lies.

1) sin 0.3614 2) tan 2.553 3) cos 0.866

43

Examples

Determine the quadrant in which the terminal side of θ lies, subject to both given conditions.

1) sin 0, cos 0 2) sec 0, cot 0

44

Examples

Find the exact value of the six trigonometric functions of θ if the terminal side of θ passes through point (3, -5).

45

The values of the trig functions for non-acute angles (Quads II, III, IV) can be found using the values of the corresponding reference angles.

Reference Angles (Sect 8.2)

Definition of Reference Angle

Let be an angle in standard position. Its reference angle is the acute angle formed by the terminal side of and the horizontal axis.

ref

46

Example

Find the reference angle for 225

Solution y

x

ref

By sketching in standard position, we see that it is a 3rd quadrant angle. To find , you would subtract 180° from 225 °.

ref

225 180

45

ref

ref

47

So what’s so great about reference angles?

Well…to find the value of the trig function of any non-acute angle, we just need to find the trig function of the reference angle and then determine whether it is positive or negative, depending upon the quadrant in which the angle lies.

For example,

1sin 225 (sin 45 )

2

45° is the ref angleIn Quad 3, sin is negative

48

Example

Give the exact value of the trig function (without using a calculator).

cos 150

49

Examples (Text p 239 #6 & 8)

Express the given trigonometric function in terms of the same function of a positive acute angle.

6) tan 91 , sec 345 8) cos 190 , cot 290

50

Now, of course you can simply use the calculator to find the value of the trig function of any angle and it will correctly return the answer with the correct sign.

Remember:

Make sure the Mode setting is set to the correct form of the angle: Radian or Degree

To find the trig functions of the reciprocal functions (csc, sec, and cot), use the button or enter [original function] .

51

Example

Evaluate . Round appropriately.

Set Mode to Degree

Enter: OR

cot 324.0

: cot 324.0 1.38ANS

52

HOWEVER, it is very important to know how to use the reference angle when we are using the inverse trig functions on the calculator to find the angle because the calculator may not directly give you the angle you want.

r-5

y

x

(-12, -5)

-12

Example: Find the value of to the nearest 0.01°

53

Examples

Find for 0 360

1) sin 0.418 2) tan 1.058

54

Examples

Find for 0 360

3) cos 0.85 4) cot 0.012, sin 0

55

BONUS PROBLEM

Find for 0 360 without using a calculator.

2sin 1 0

56

SUPER DUPER BONUS PROBLEM

Find for 0 360 without using a calculator.

24(sin ) 9 6

57

Trig functions of Quadrantal Angles

To find the sine, cosine, tangent, etc. of angles whose terminal

side falls on one of the

axes , we will use the circle.

(..., 180 , 90 , 0 , 90 , 180 , 270 , 360 ,...)

(0, 1) 90

(1, 0)(-1, 0)

(0, -1)

0

270

180

Unit Circle:

Center (0, 0)

radius = 1

x2 + y2 = 1

58

Now using the definitions of the trig functions with r = 1, we have:

sin csc1

cos sec1

tan cot

1

1

yy

x

y y r

r y

xy x

x

x

y

x r

r x

59

Find the value of the six trig functions for

Example

90

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

0

270

90

180

sin 901

cos 901

tan 90

1csc 90

1sec 90

cot 90

y y

rx x

ry

xr

y y

r

x xx

y

60

Find the value of the six trig functions for

Example

0

sin 0

cos 0

tan 0

1csc 0

1sec 0

cot 0

y

x

y

x

y

xx

y

61

Find the value of the six trig functions for

Example

540

sin 540

cos 540

tan 540

1csc 540

1sec 540

cot 540

y

x

y

x

y

xx

y

62

In general, for in radians,

A second way to measure angles is in radians.

Radian Measure (Sect 8.3)

s

r

Definition of Radian:

One radian is the measure of a central angle that intercepts arc s equal in length to the radius r of the circle.

63

Radian Measure

2 radians corresponds to 360

radians corresponds to 180

radians corresponds to 902

2 6.28

3.14

1.572

64

Radian Measure

65

Conversions Between Degrees and Radians

1. To convert degrees to radians, multiply degrees by

2. To convert radians to degrees, multiply radians by

180

180

Example

Convert from degrees to radians: 210º

210

66

Conversions Between Degrees and Radians

Example

a) Convert from radians to degrees:

b) Convert from radians to degrees: 3.8

3

4

3

4

3.8

67

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):

d) Convert from radians to degrees: 13

6

13

6

675

675

68

Conversions Between Degrees and Radians

Again!

e) Convert from degrees to radians (to 3 decimal places):

f) Convert from radians to degrees (to nearest tenth): 1 rad

5252

1

69

Examples

Find to 4 sig digits for 0 2

sin 0.9540