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