Vectors Dec. 11

What we did in class

1. 
   Wrapped up Law of Sines and Law of Cosines with problems 1-7 on pg 567 in class



2. Introduced vectors (Not on final exam)




3. Rewiewed for the quiz (Dec. 12)

Vectors: Have magnitude and direction

Examples of vectors are velocity, force, acceleration, etc.

Vectors are named with a letter with an arrow over the top or as a bold letter like our book uses

Vectors can be moved anywhere as long as its direction and magnitude remain the same.

When you add vectors, you connect them tip to tail

Vectors can also be put onto a graph.
To slove vectors on a graph, you can use the distance formula or make it look like a triangle and solve it using the unit circle.

When you are dealing with vectors that have names at their start and at the finish, it matters which letter comes first when you name it. AB is the same as BA only they travel in opposite directions.

Vectors with two letters assigned to them are written inside absolute value bars. Sometimes the bars are double thick. This shows that they have magnitude.

Final Questions

Ch. 1 Pg# 130, 19-21, 25-28, 33-52, 79-86
Ch. 2 Pg# 207, 1-38, 47-73, 85-94, 103, 104, 107, 108
Ch. 6 Pg# 479, 17-44, 51-68, 71-74, 85-88, 95-98
Ch. 7 Pg# 520, 1-21 (These are difficult), 29-48, 55
Ch. 8 Pg# 567, 1-8, 11-14, 16

Bold problems are good to study for the Quiz Wednesday

The Quiz is on:
sum and difference identities
double and half identities
equations of identities
law of sine and cosine

EQUATIONS

Example #1:
sin2x=sinx [0,2pi]
2sinxcosx=sinx
2sinxcosx-sinx=0
sinx(2cosx-1)=0
Solve by setting terms equal to 0.
sinx=0 2cosx-1=0
x=0, 2pi, pi cosx=1/2
x=pi/3, 5pi/3

Example #2:
sin2x+cos3x=0 [0, 2pi]
2sinxcosx+cos(2x+x)=0
2sinxcosx+cos2xcosx-sin2xsinx=0
2sinxcosx+cos2xcosx-2sinxcosxsinx=0
2sinxcosx+(1-2sin²x)cosx-2sin²xcosx=0
2sinxcosx+cosx-2sin²xcosx-2sin²xcosx=0
2sinxcosx+cosx-4sin²xcosx=0
cosx(2sinx+1-4sin²x)=0
cosx=0

x=0, pi/2, 3pi/2

-4sin²x+2sinx+1=0
Use quadratic formula to get answer for sinx=
Use inverse sign to find remaining 4 x values
5.97, 3.45, .942, 2.19

Example #3: (Proof)
sin4x=2sin2xcos2x
sin2x=2sinxcosx
=sin2(2x)
sin4x=sin4x
QED

Homework: pg 510 #35, 36 #51-57 Odd
Pg 518 # 3-33 multiples of 3

More on Solving Trig Equations

Homework: page 502 #9-31 odd

Sorry this post is SO late, I forgot about it over Thanksgiving and unfortunately, I am just now posting. SO SORRY! Also, sorry the sizes are so large, I did not know how to adjust that easily :P

Using Law of Sines and Law of Cosines

When do you use Law of Sines and when do you use Law of Cosines?

Well,

Law of Sines

more

AAS

SSA (ambiguous)

Law of Cosines

more sides given

SAS

SSS

Ambiguous Case: SSA

There are five possible combinations of triangles when you are given two sides and an angle.



In Case 1, you can see that side b >a

In Case 2, b forms a single right angle triangle with c. Here b equals h, h being the height of the triangle, and yields a single right angle triangle.

In Case 3, a>b>h, forming two triangles. Side b is too long to form a single right angle triangle, but yet is also too short to swing out farther than side a which would result in only one triangle. Instead it forms to triangles, one acute triangle and one obtuse triangle.

In Case 4, side b is equal side a, resulting in a single isosceles triangle. Being an isosceles triangle, angle A and angle B are also equal. Side b cannot be placed anywhere else or it would not form a triangle.

In Case 5, side b is > side a. It forms one triangle only, with side b stretching out opposite of a. It cannot be on the other side of a because then it would not form a triangle.

Ok here is how we use the Law of Sines to solve a triangle.

Lets say we are given:
a = 21
b=20

Lets start by solving for
(sinA)/a = (sinB)/b

So we plug in the numbers that we have:

(sin 33)/21 = (sinB)/20
20(0.545)/21=sinB
0.519=sinB
B=31.268°

We have found
180-33-31.268=115.732°

We then can find c with the Law of Cosines:

c(c)=a(a)+b(b)-2abcosC
c(c)=441+400-2(21)(20)cos115.732
c(c)=1205.696
c=34.723

Sum and Difference Identities

sin (x + π) ≠ sin x + sin π Cannot distribute Cosine of a difference (proof) cos (u-v) = cos u cos v + sin u sin v

Identities

Sine of a Sum

sin (u+v) = sin u cos v + cos u sin v

Sine of a Difference

sin (u-v) = sin u cos v - cos u sin v

Cosine of a Sum

cos (u+v) = cos u cos v - sin u sin v

Cosine of a Difference

cos (u-v) = cos u cos v + sin u sin v

Tangent of a Sum

tan (u+v) = (tan u + tan v)/(1-tan u tan v)

Tangent of a Difference

tan (u-v) = (tan u - tan v)/(1+ tan u tan v)

When to use it:

Example:

cos(15) = cos (45-30)

Simplifying Expressions
Example: x +1 + x -3
X2-4x+4 x -2
= x +1 + x -3
x2-4x+4 x -2

1) Factor the denominator and then find a common denominator.
= x+1 + (x-3) (x-2)
(x-2)(x-2) (x-2) (x-2)
2) Combine the fractions
= x+1+ (x-3)(x-2)
(x-2)(x-2)
3) Simplify
= 2x-2
x-2



Example: cosx – sinx
1-sinx cosx
= cosx – sinx
1-sinx cosx
1) Find a common denominator for both fractions
= cosx (cosx) – sinx (1-sinx)
(1-sinx)(cosx) cosx (1-sinx)
2) Combine the fractions
= cos2x – sinx(1-sinx)
(1-sinx)(cosx)
3) Simplify
= cos2x – sinx+sin2x
(1-sinx)(cosx)
4) Use the identity sin2x+cos2x = 1 in the numerator.
= 1-sinx
(1-sinx)(cosx)
5) Simplify
= 1
cosx
6) Use the reciprocal identity
= secx



Factoring
Example: 1 + cosx - sin2x
= 1 + cosx - sin2x
1) Use the identity sin2x + cos2x = 1 to substitute sin2x for (1-cos2x).
= 1 + cosx – (1 - cos2x)
2) Distribute the negative sign into the parenthesis.
= 1 + cosx – 1 + cos2x
3) Simplify
= cosx + cos2x
= cosx(1 + cosx)









Example: sec2x + tanx - 3
= sec2x + tanx - 3
1) Use the identity 1+tan2x = sec2x
= 1 + tan2x + tanx – 3
2) Simplify
= tan2x + tanx – 2
3) Factor
= (tanx + 2)(tanx – 1)




You can check if two expressions are equivalent by using your graphing calculator. Graph the two expressions, but change one of the expressions to the bouncing ball.

Homework: pg 487 # 1-7 odd, 15-43 odd

Solving Trig Equations - 11.12.07

Data/Trig Functions

sine curve regression


make sure your calculator is in radian mode

calculator:

1) stat button-enter data
2) 2nd/stat plot
3) move to dot plot-hit enter
4) set window
5) view graph


sine curve function

after creating the sine curve...

1) stat
2) move to calc-SinReg
3) enter: L1, L2, (commas are important)
4) vars-move to y-vars
5) function
6) enter Y1
7) hit enter

you should see something like:

y=
a=
b=
c=
d=

the data will also be a y= function in Y1

to check other types of curves for fit:

calculator:

1) stat-calc
2) choose a regression-CubReg, QuartReg, etc.-enter
3) enter: L1, L2,
4) vars
5) y-vars-function
6) enter Y2-hit enter
7) view graph



Quiz-Tuesday, November 6

Topics:

1) graphing sine/ cosine by hand
2) identifying graphs of tan/cot/sec/csc
3) unit circle
4) inverse trig
5) application

Inverse Trig

sin -1 = inverse sine
cos -1 = inverse cosine
tan -1 = inverse tangent

circular functions {arc cos, arc sin, and arc tan} arc is measured by the angel taht opens up to it.

Inverse undoes the function
x2 & Square root of x

multiplicative inverse = 3 times 1/3 = 1
additive inverse = 3 + -3 = 0

f(x) = x2
g(x) = square root of x
(f0g)(x) = (g0f)(x) = x
x is greater than or equal to zero.

Inverses

• reflections over y = x
  
• (x, y) - (y, x)


• (f0g)(x) = (g0f)(x) =x
  

To take an inverse you have to first restrict the domain

For inverses inputs and outputs switch.

Inverse Sine

y=sin(x)

Domain: (-3.14/2, 3.14/2)

Range: (-1, 1)

y=sin-1(x)


Domain: (-1, 1)


Range (-3.14/2, 3.14/2)

Inverse Cosine

y=cos(x)

Domain: (0, 3.14)

Range: (-1, 1)














y=cos-1(x)

Domain: (-1,1)

Range: (0, 3.14)

Inverse Tangent

y=tan(x)

Domain: (-3.14/2, 3.14/2)

Range: (-infinity, infinity)

y=tan-1(x)

Domain: (-infinity, infinity)

Range: (-3.14/2, 3.14/2)

Graphs of Other Trig Functions

Graphing Trig Functions 10/12

*other graphs will change from these two original graphs

*basic form

A sin (or cos) b (x-c) + d

Notes- Graphing Trig Functions

1. amplitude

2. period (P = 2∏/B)

3. increment (P/4)

4. vertical translation (midline)

5. horizontal translation (starting point)

Examples:

f(x) = 3cos (x/2 - ∏/6) + 1

doesn’t look like basic form – so factor out ½

f(x) = 3cos ½ (x - ∏/3) + 1

amplitude= 3

period= 2∏/B = 2∏/ ½ = 4∏

increment= 4∏/ 4 = ∏

vertical translation = up one unit (midline)

horizontal translation = right ∏/3 (starting point)

*when graphing scale based on the increment

Y= -2 sin ∏/3 (x - 2) + 1

Amplitude = 2 (the negative cause the graph to begin down rather than upward )

Period = 2∏/ ∏/3 = 6

Increment= 6/4 or 1.5

Vertical= up one unit

Horizontal right two units

Homework

Page 439 in book, #’s 45-52

Do by hand then check with your calculator

Review for quiz (optional)

Page 708; 68-74, 85-94, 99, 100, 107, 108

Page 479: 1-48, 51-54

What you need to know:

Functions: operations on functions, composition, inverses and transformations

Trig: right triangle trig, and unit circle

*there will be a non-calculator portion

General Graphs of Sine and Cosine

Radian: 1 radian is the length of 1 radius bend around the arch.
1 radian ≈ 57.29°

Sinusoids y=a sin b(x-c)+d, y=a cos b(x-c)+d

d- vertical translation
c- horizontal translation (phase shift)
a- amplitude (vertical stretch) if negative it reflects over X axy
b- horizontal strectch/shrink (frequency)

** ∏=pie (3.14)


Sine
» Domain: (-∞,∞)
» Range: [1,-1]
» Period: 2∏
» Zeros: ∏n n=any integer


Cosine
» Domain: (-∞,∞)
» Range: [1,-1]
» Period: 2∏
» Zeros: ∏/2 + ∏n n=any integer



ex: y=3cos 2(x-∏/4)+ 1

period: 2∏/2 = ∏
incrament: ∏ /4
a: 3
horizontal shift: ∏ /4
vertical shift: 1

just a note

in case you guys haven't already tried this, if you click on one of the images in the blog it will pull it up full size, which should make it easier to read

Why Does Trig Work? [10/2/07]

Why Does Trig Work?



Triangle Problem:

1.Draw a Trianlge

2.Measure easch side to the nearest 10th of a centimenter

3.Solve for:



AB/AC BC/AC AB/BC

Class Results:

-The actual results are AB/AC = .8192 BC/AC = .5736 AB/BC = 1.4281

It is important to recognize that the ration is constant and that similarity is why trig works

Bearing:

Right Triangle Trig:

SOH CAH TOA :

sin: opp/hyp cos: adj/hyp tan: opp/adj


Reciprocal relationships:

csc: hyp/opp or 1/sin of angle sec: hyp/adj or 1/cos of angle

cot: adj/opp or 1/tan of angle

Ex. :

The sine of an angle equals 5/6 Find values of all 6 trig functions.

Start by using the pythagorean theorem

5^2 + x^2 = 6^2

25 + x^2 = 36

x^2 = 11

x = the square root of 11

Then use SOH CAH TOA to solve for the other 5 trig functions

cos : square root of 11/ 6 sec: 6/square root of 11

tan: 5/square root of 11 cot: square root of 11/ 5

csc: 6/5

Ex. :

Solving a Triangle: Finding all missing pieces

180 - 143 = 37

sin 37 = b/8

b = 8 x sin 37

b= 4.8

cos 37 = a/8

a = 8 x cos 37

a = 6.4


Angle of Depression

30-60-90 Triangle

45-45-90 Triangle

Homework:

pg. 399 1-23 odd 41-55 odd (given on the friday before)

pg. 399 57-73

pg.411 1-23 odd 51-57 odd 71-85 odd


Just for fun, one of my favorite quotes is:

"Twenty years from now you will be more disappointed by the things you didn't do than by the things you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover." ~ Mark Twain

Extending Trig Functions [10/05/07]

r= Radius

r= √x²+y²

SINΘ= y/r

COSΘ= x/r

TANΘ= y/x

CSCΘ= r/y

SECθ= r/x

COTθ= x/y

*Obviously, x,y,r≠0 because 1) you can't divide by 0, and 2) you can't have a 0° angle in a triangle because, well, then it wouldn't be triangle, would it?

Example:

P (-2,3) Find all six trig functions.

r= √2²+3²
r= √4+9
r= √13

SINΘ= 3/√13

COSΘ= -2/√13

TANΘ= 3/-2

CSCΘ= √13/3

SECΘ= √13/-2

COTΘ= √-2/3

*At this point, Truitt asked, "What exactly are we finding with the functions?"

Jenna answered, "We find theta (θ)."

Marchetti enlightened us further.

Quadrant I= All positive

Quadrant II= SIN +

Quadrant III= TAN+

Quandrant IV= COS+

*The reciprocal functions will be positive at the same time their original functions are.

*"All Star Trig Class"

...A: all positive in quadrant I, S: SIN positive in quadrant II, T: TAN positive in quadrant III, C: COS positive in quadrant IV.

Quadrantal Angles:

→Big word for "angle that takes up entire quadrant"
→Class nicknamed quadrantal angles 'Steve' for some reason...

→Angles begin and end on any axis

Unit circle: r=1

So...

Reference Angles:

→An angle formed by the terminal side of an angle in standard position and the horizontal (x) axis.
→Are our friends.

Homework:

→Unit Circle handout
→Revisions
→p424: 1-55 odd