Download - Chapter 5: Analytic Trigonometry
Chapter 5: Analytic Trigonometry
Section 5.1a: Fundamental IdentitiesHW: p. 451-452 1-7 odd,
27-49 odd
Is this statement true?2 1 11
x xx
This identity is a true sentence, but onlywith the qualification that x must be in thedomain of both expressions.
tan sin cos
If either side of the equality is undefined (i.e., at x = –1), thenthe entire expression is meaningless!!!
The statement is a trigonometric identitybecause it is true for all values of the variable for which bothsides of the equation are defined.
The set of all such values is called the domain of validity ofthe identity.
Basic Trigonometric Identities
1cscsin
Reciprocal Identities1seccos
1cottan
1sincsc
1cossec
1tancot
sintancos
Quotient Identitiescoscotsin
0 is in the domain of validity of exactly three of the basicidentities. Which three?
Basic Trigonometric Identities
1cscsin
Reciprocal Identities1seccos
1cottan
1sincsc
1cossec
1tancot
sintancos
Quotient Identitiescoscotsin
0 For exactly two of the basic identities, one side of the equationis defined at and the other side is not. Which two?
Basic Trigonometric Identities
1cscsin
Reciprocal Identities1seccos
1cottan
1sincsc
1cossec
1tancot
sintancos
Quotient Identitiescoscotsin
0 For exactly three of the basic identities, both sides of theequation are undefined at . Which three?
Pythagorean Identities
cos ,sinP t t
2cos tDivide by :
Recall our unit circle:
(1,0)
PWhat are the coordinates of P?
cost
sint
2 2cos sin 1t t So by the Pythagorean Theorem:
2 2
2 2 2
cos sin 1cos cos cos
t t
t t t 2 21 tan sect t
Pythagorean Identities
cos ,sinP t t
2sin tDivide by :
Recall our unit circle:
(1,0)
PWhat are the coordinates of P?
cost
sint
2 2cos sin 1t t So by the Pythagorean Theorem:
2 2
2 2 2
cos sin 1sin sin sin
t t
t t t 2 2cot 1 csct t
Pythagorean Identities
tan 5 Given and , find and .
2 2cos sin 1 2 21 tan sec 2 2cot 1 csc
cos 0
We only take the positive answer…why?
sin cos2 2sec 1 tan 21 5 26 sec 26
1cos26
tan 5 sin 5cos
sin 5cos
1sin 526
5sin26
Cofunction Identities
sin cos2
cos sin
2
tan cot2
cot tan
2
sec csc2
csc sec
2
Can you explain why each of these is true???
Odd-Even Identities
sin sinx x cos cosx x
csc cscx x sec secx x
tan tanx x cot cotx x
If , find .cos 0.34
Sine is odd
sin 2
sin sin2 2
Cofunction Identity cos 0.34
Simplifying Trigonometric Expressions
3 2sin sin cosx x xSimplify the given expression.
How can we support this answer graphically???
2 2sin sin cosx x x
sin 1xsin x
Simplifying Trigonometric Expressions
2
sec 1 sec 1sin
x xx
Simplify the given expression.
Graphical support?
2
2
sec 1sinxx
2
2
tansin
xx
2
2 2
sin 1cos sin
xx x
2
1cos x
2sec x
Simplifying Trigonometric Expressions
1 tan1 cot
xx
Simplify the given expressions to either a constant or a basictrigonometric function. Support your result graphically.
sin1coscos1sin
xxxx
cos sincos
sin cossin
x xx
x xx
cos sin sincos sin cosx x xx x x
sincosxx
tan x
Simplifying Trigonometric ExpressionsSimplify the given expressions to either a constant or a basictrigonometric function. Support your result graphically.
2 2
2 2
sec tancos sin
u uv v
11
1
Simplifying Trigonometric Expressions
sec tan sec tansec
y y y yy
Use the basic identities to change the given expressions to onesinvolving only sines and cosines. Then simplify to a basictrigonometric function.
1 sin 1 sincos cos cos cos
1cos
y yy y y y
y
1 sin 1 sin coscos cos 1
y y yy y
Simplifying Trigonometric ExpressionsUse the basic identities to change the given expressions to onesinvolving only sines and cosines. Then simplify to a basictrigonometric function.
1 sin 1 sin coscos cos 1
y y yy y
21 sin sin sin
cosy y y
y
21 sin
cosyy
2coscos
yy
cos y
Simplifying Trigonometric ExpressionsUse the basic identities to change the given expressions to onesinvolving only sines and cosines. Then simplify to a basictrigonometric function.
2
2 2
sec cscsec csc
x xx x
2
2 2
1 1cos sin1 1
cos sin
x x
x x
2 2
2 2 2
1 cos sincos sin sin cos
x xx x x x
2 2
sinsin cos
xx x
sin1x
sin x
Let’s start with a practice problem…Simplify the expression
cos sin1 sin cos
x xx x
cos cos sin 1 sin1 sin cos cos 1 sin
x x x xx x x x
cos cos sin 1 sin1 sin cos
x x x xx x
2 2cos sin sin1 sin cosx x x
x x
1 sin
1 sin cosx
x x
sec x1cos x
How about somegraphical support?
Combine the fractions and simplify to a multiple of a power of abasic trigonometric function.
1 11 sin 1 sinx x
1 sin 1 sin
1 sin 1 sin 1 sin 1 sinx x
x x x x
1 sin 1 sin1 sin 1 sin
x xx x
2
21 sin x
22sec x
2
2cos x
Combine the fractions and simplify to a multiple of a power of abasic trigonometric function.
1 1sec 1 sec 1x x
sec 1 sec 1sec 1 sec 1x xx x
2
2sec 1x
2
2tan x
22cot x
Combine the fractions and simplify to a multiple of a power of abasic trigonometric function.sin 1 cos1 cos sin
x xx x
22sin 1 cossin 1 cosx xx x
2 2sin cos 2cos 1sin 1 cosx x x
x x
2sin x
2csc x
2 2cos
sin 1 cosx
x x
2 1 cossin 1 cos
xx x
Quick check of your algebra skills!!!
212 8 15x x 8b180a c
10,18
Factor the following expression (without any guessing!!!)
What two numbers have a product of –180 and a sum of 8?
Rewrite middle term: 212 10 18 15x x x Group terms and factor: 2 6 5x x 3 6 5x Divide out common factor: 6 5 2 3x x
Write each expression in factored form as an algebraicexpression of a single trigonometric function.
2sin 3 sin 1x x 2sin 2sin 1x x
e.g.,
sinu xLet
Substitute: 2 2sin 2sin 1 2 1x x u u
Factor: 21u
“Re”substitute for your answer: 2sin 1x
Write each expression in factored form as an algebraicexpression of a single trigonometric function.
2sin 3 sin 1x x
221 cossec
xx
e.g.,
21 2cos cosx x
21 cos x
Write each expression in factored form as an algebraicexpression of a single trigonometric function.
2sin 3 sin 1x x
2sin cos 1x x
e.g.,
2sin 1 sin 1x x
sin 1 sin 2x x
2sin sin 2x x sinu xLet 2 2u u 1 2u u
Write each expression in factored form as an algebraicexpression of a single trigonometric function.
2sin 3 sin 1x x
2 2sec sec tanx x x
e.g.,
2sec 1 sec 1x x
2 2sec sec sec 1x x x 22sec sec 1x x
Write each expression as an algebraic expression of a singletrigonometric function. 2sin 3x
2tan 11 tan
e.g.,
tan 1 tan 11 tan
tan 1
2tansec 1
xx
2sec 1sec 1
xx
sec 1x
sec 1 sec 1sec 1x x
x