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

Notes from 2.4: Operations on Functions/ Composition of Functions

*Sorry these notes weren't up sooner, but I posted them before, and then they somehow didn't show up on the blog. Sorry guys*

1. Composition of a Function:
-(fog)(x) or “f circle g of x” aka f(g(x))
--f(x)=x2–5
--g(x)=3x-4
(fog)(x)=f(g(x))
=f(3x-4)
=(3x-4)2-5
=9x2-24x+16-5
(fog)(x)=9x2-24x+11
(gof)(x)=g(f(x))
=g(x2–5)
=3(x2–5)-4
=3x2-15-4
(gof)(x)=3x2-19
2. Inverses:
- use PEMDAS in reverse (SADMEP)
-inverses are always functions
-1/2 of inverse (quadratics) will show because the other half does not pass the vertical line test. (Seen in graphing on calculator)
--f(x)=2x-1
f-1(x)=
g(x)=(x-3)2
g-1(x)=
or:
g-1(x)= +3
3. Graphing Inverses:
-inverse will be a reflection of equation over the y=x line
-points switch from (x,y) to (y,x)

-graph vs. y=x

4. Algebra of Inverses:
--f(x)=
f-1(x) » y=
y=
x= (x switches places with y)
x2=y-3
y=x2+3 » f-1(x)= x2+3
D , D-1:
5. Algebra Cont. Checking for Inverses:
--f(x)=x3+1
--g(x)=
--(fog)(x)= (gof)(x)=x
f(g(x)): g(f(x)):
f( ) g(x3+1)
( 3+1 ) -1
x-1+1=x ) =x YES, they are inverses!
Homework (due Sept. 28) - 180:15-35 odd, 197: 1-15 odd, 33-41 odd, 71-80
Quiz (Sept. 26) – Quadratics, Solving with Calculator, Absolute Value, Inequalities, Application Problem

Notes 9/28/2007