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