DIFFERENCE AND ADDITION IDENTITIES

MORNING NOTES


This is called a co function property for circular functions.

Which can be written like this:

sin (pi/2 - x) = cosx or cos (pi/2 - x) = sinx

cot (pi/2 - x) = tanx or tan (pi/2 - x) = cotx

csc (pi/2 - x) = secx or sec (pi/2 - x) = cscx



ex) Can you find the value for cos15? (W/out your calculator)

-you should know 0, 30, 45, 60, 90, 180, 270.
Using these values we’ll get to cos15

Which values if you subtract or add you’ll get 15?
Cos(45-30)



coordinates for d1
p1 - coordinates of p(α) (cosα , sinα)
p2 - coordinates of p(β) (cosβ , sinβ)

coordinates for d2
p3 - coordinates of p(α-β) (cos(α-β) , sin(α-β)
p4 - coordinates of p(P0) (1,0)

we will be using the distance formula
d= (square root)(x2-x1)^2 + (y2-y1)^2

d=(square root)(cosβ- cosα)^2 + (sinβ-sinα)^2 = (square root)(cos(α-β)-1)^2 + (sin(α-β)-0)^2


(root)(cos^2β-2cosβcosα+cos^2α + sin^2β-2sinβsinα+sin^2α
(root)(cos^2β+sin^2β+cos^2α+sin^2α-2cosβcosα-2sinβsinα
(root)(1)(1) - 2cosβcosα-2sinβsinα
((root)2-2cosβcosα-2sinβsinα)^2 <----we square it to get rid of the root sign

=

(root)cos^2(α-β)-2cos(α-β)+1 + sin^2(α-β)
(root)cos^2(α-β)+sin^2(α-β)-2cos(α-β)+1
(root)(1) + (1) - 2cos(α-β)
((root)2-2cos(α-β))^2 <----we square it to get rid of the root sign


we do this because we're trying to get to a point where it's cos(α-β)

2-2cosβcosα-2sinβsinα = 2-2cos(α-β) <--2's cancel
-2cosβcosα-2sinβsinα/-2 = -2cos(α-β)/-2 <--divide both sides by -2 to get rid of the denominator

and you get

cosβcosα+sinβsinα = cos(α-β) <--this is called difference identity


as you can remember from beofre we were trying to look for cos15.
we can also do this cos(45-30) <--45 is the α and 30 is β

cos(45-30) = cos45cos30 + sin45sin30
=(root2/2)(root3/2) + (root2/2)(1/2)
=(root6/4) + (root2/4)
=root6 + root2/4

now addition....

cos(105) **it doesnt matter whether you multiply (α)(β) or (β)(α) <--it's still the same thing

so...
cos(105)
what can you add to get 105 from the numbers : 0, 30, 45, 60, 90, 180, 270 ?
60 and 45 . .riiight?

cos(60+45)
cos(α+β) = cos(α-(-β))
-------- =cosαcos(-β) + sinαsin(-β)
-------- =cosαcosβ - sinαsinβ


we can say that cos(-β)=cosβ

so the addition identity is written like this...
cos(α+β) = cosαcosβ-sinαsinβ

AFTERNOON NOTES

use identities to find the exact value for 7pi/12 and cos 7pi/12

we know : 0 , pi/6 , pi/3 , pi/4 , pi/2 , pi , 3pi/2 , 2pi

pi/3 = 4/12
pi/4 = 3pi/12 . . . we get 7pi/12



sin(pi/3 + pi/4) = sin(pi/3)cos(pi/4) + cos(pi/3)sin(pi/4)
---------------- =(root3/2)(root2/2) + (1/2)(root2/2)
---------------- =(root6/4) + (root2/4)
---------------- =(root6 + root2)/4

cos(α+β)= cosαcosβ - sinαsinβ
cos(pi/3 + pi/4)= cos(pi/3)cos(pi/4) - sin(pi/3)sin(pi/4)
----------------= (1/2)(root2/2) - (root3/2)(root2/2)
----------------= (root2/4) - (root6/4)
----------------= (root2 - root6)/4

yess! finally im done. hopefully you all understand. this took pretty long but i had fun with it. so the next scribe i choose well not really is nikka-elle... good luck nicky :)