Post on 18-Dec-2015
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SAT Multiple Choice Question(s)
4 cm
6 cm The figure above shows how a
rectangular piece of paper is rolled into the shape of a cylinder. If it is assumed that the 4-centimeter sides of the rectangle meet with no overlap, what is the area, in square centimeters, of the base of the cylinder?
(a)
(b)
(c)
(d)
(e)
916
9
4
4
Essential Question: How do I use trig identities to solve
equations and verify identities?
How do I use fundamental identities to verify other identities?
Reciprocal Identities
sincsc
1
sincsc
22
1
cossec
1
tancot
1
cscsin
1
seccos
1
cottan
1
Also work with powers…
See pg 454
Quotient Identities
tansin
cos
cot
cos
sin
generating the …Pythagorean Identities
(cos , sin )
cos
sin 1
a2 + b2 = c2
(cos )2 + (sin )2 = 12
cos2 + sin2 = 1
cos2 means the same thing as (cos )2
generating the…Pythagorean Identities
cos2 + sin2 = 1cos2 cos2 cos2
1 + tan2 = sec2
generating the…Pythagorean Identities
cos2 + sin2 = 1sin2 sin2 sin2
+ 1cot2 = csc2
Pythagorean Identities
cos2 + sin2 = 1
1 + tan2 = sec2
+ 1cot2 = csc2
You can also manipulate them…
These are very important!
manipulating the Pythagorean Identities
cos2 + sin2 = 1 - cos2 - cos2
sin2 = 1 - cos2
cos2 + sin2 = 1 - sin2 - sin2
cos2 = 1 - sin2
Ex. Simplify xxx cotcoscsc
Ex. Simplify xxx tancsccos
1.) Work with one side of the equation. (The complicated side first).
2.) Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator.
Guidelines for verifying… pg 462
3.) Look for opportunities to use the identities.
4.) If the preceding guidelines do not help, try converting all terms to sines and cosines.
Guidelines for verifying…
5.) Try something! Even making an attempt that leads to a dead end gives insight.
pg 462
factor out a GCF
Ex. Verify sin - cos2 sin =sin
= sin (1 - cos2 )
= sin (sin2 )
= sin3 Goal: Single Trig Function, if possible
Substitute w/ Pythag ID
Multiply
3
2 2 2sin sin tanx x x2 2sin (1 tan )x x
sin (sec )2 2x x
sincos
22
1x
x2
2
sin
cos
x
x
2tan x
Ex #2b Verify you try…
2tan x
2tan x
22
2
sec 1sin .
secVerify theidentity
2sin
tan
sec
2
2
22
sec
1tan
22 costan
2
2
2
coscos
sin
Ex.
sin2
2
2
sec 1
sec
2sin
2sin
Another way to do #1…
sec
sec sec
2
2 2
1
1 2 cos
sin2
Write in terms of sin or cos
cossin cos
sin
tt t
t
2cossin
sin
tt
t
sin
sin
t
t2 2sin cos
sin
t t
t
1
sin t csc t
Ex. Verify
sin(t) + cot(t) cos(t)= csc(t)
Multiply
Add, common denominator
Substitute
Substitute
1
1 sin x
1
1
sin
sin
x
x
2
1 sin
1 sin
x
x
2
1 sin
cos
x
x
2 2
1 sin
cos cos
x
x x
xx
x
x cos
1
cos
sin
cos
12
2sec tan secx x x
Ex. Verify
Multiply by the conjugate
21sec tan sec
1 sinx x x
x
What are your questions?