Download chapter 5 - James Bac Dang

CHAPTER 5
ANALYTIC TRIGONOMETRY
5.1 Verifying Trigonometric
Identities
• Objectives
– Use the fundamental trigonometric identities
to verify identities.
Fundamental Identities
• Reciprocal identities
csc x = 1/sin x sec x = 1/cos x cot x = 1/tan x
• Quotient identities
tan x = (sin x)/(cos x) cot x = (cos x)/(sin x)
• Pythagorean identities
sin 2 x  cos 2 x  1
tan 2 x  1  sec 2 x
cot x  1  csc x
2
2
Fundamental Identities
(continued)
• Even-Odd Identities
– Values and relationships come from
examining the unit circle
sin(-x)= - sin x
cos(-x) = cos x
tan(-x)= - tan x
cot(-x) = - cot x
sec(-x)= sec x
csc(-x) = - csc x
Given those fundamental identities,
you PROVE other identities
• Strategies: 1) switch into sin x & cos x, 2)
use factoring, 3) switch functions of
negative values to functions of positive
values, 4) work with just one side of the
equation to change it to look like the other
side, and 5) work with both sides to
change them to both equal the same thing.
• Different identities require different
strategies! Be prepared to use a variety of
techniques.
Verify:
1  cos x
(csc x  cot x) 
1  cos x
2
• Manipulate right to look like left. Expand the
binomial and express in terms of sin & cos
(csc x  cot x) 2  csc 2 x  2 csc x cot x  cot 2 x
1
2 cot x cos 2 x 1  2 cot x  cos 2 x





2
2
sin x sin x sin x sin x
sin 2 x
(1  cos x) 2 (1  cos x) 2 (1  cos x)(1  cos x) 1  cos x




2
2
sin x
1  cos x
(1  cos x)(1  cos x) 1  cos x
5.2 Sum & Difference
Formulas
• Objectives
– Use the formula for the cosine of the
difference of 2 angles
– Use sum & difference formulas for cosines &
sines
– Use sum & difference formulas for tangents
cos(A-B) = cosAcosB + sinAsinB
cos(A+B) = cosAcosB - sinAsinB
• Use difference formula to find cos(165 degrees)
cos(165)  cos( 210  45)
 cos 210 cos 45  sin 210 sin 45
1 2
2  2
6  2 2 6


 3



2 2
2
4
2
4
sin(A+B) = sinAcosB + cosAsinB
sin(A-B) = sinAcosB - cosAsinB
13
find (sin
)
12
13
 9 4 
 3  
sin
 sin     sin   
12
 12 12 
 4 3
3 
3 
2 1  2 3
2 6
 sin cos  cos sin   
 
4
3
4
3 2 2 2 2
4
tan A  tan B
tan( A  B) 
1  tan A tan B
tan A  tan B
tan( A  B) 
1  tan A tan B
5
3 2
Find (tan )  tan(

)
12
12 12


3
tan  tan
1
 
4
6 
3
 tan(  ) 
4 6 1  tan  tan 
3
1  1
4
6
3
3 3

3 3
5.3 Double-Angle, PowerReducing, & Half-Angle Formulas
• Objectives
– Use the double-angle formulas
– Use the power-reducing formulas
– Use the half-angle formulas
Double Angle Formulas
(developed from sum formulas)
sin 2 A  2 sin A cos A
cos 2 A  cos A  sin A
2 tan A
tan 2 A 
2
1  tan A
2
2
You use these identities to find exact
values of trig functions of “nonspecial” angles and to verify other
identities.
Double-angle formula for cosine
can be expressed in other ways
cos 2 A  cos A  sin A  (1  sin A)  sin A
2
2
2
2
 1  2 sin A
2
cos 2 A  cos A  sin A  cos A  (1  cos A)
2
 2 cos A  1
2
2
2
2
We can now develop the PowerReducing Formulas.
Re call : cos 2 A  2 cos A  1,
2
cos 2 A  1  2 cos 2 A
cos 2 A  1
2
 cos A
2
2
Re call : cos 2 A  1  2 sin A
2 sin 2 A  1  cos 2 A
1  cos 2 A
2
sin A 
2
1  cos 2 A
2
thus : tan A 
1  cos 2 A
These formulas will prove very
useful in Calculus.
• What about for now?
• We now have MORE formulas to use, in
addition to the fundamental identities,
when we are verifying additional identities.
Half-angle identities are an
extension of the double-angle
ones.
 x
Re call : 2   x
 2
 x
1  cos 2 
2  1  cos x

2 x 
sin   

2
2
 2
1  cos x
 x
 x
sin    sin    
2
 2
 2
2
Half-angle identities for tangent
A 1  cos A
tan 
2
sin A
A
sin A
tan 
2 1  cos A
5.4 Product-to-Sum & Sum-toProduct Formulas
• Objectives
–Use the product-to-sum formulas
–Use the sum-to-product formulas
Product to Sum Formulas
1
sin A sin B  [cos( A  B)  cos( A  B)]
2
1
cos A cos B  [cos( A  B)  cos( A  B)]
2
1
sin A cos B  [sin( A  B)  sin( A  B)]
2
1
cos A sin B  [sin( A  B)  sin( A  B)]
2
Sum-to-Product Formulas
A B
A B
sin A  sin B  2 sin
cos
2
2
A B
A B
sin A  sin B  2 sin
cos
2
2
A B
A B
cos A  cos B  2 cos
cos
2
2
A B
A B
cos A  cos B  2 sin
sin
2
2
5.5 Trigonometric Equations
• Objectives
– Find all solutions of a trig equation
– Solve equations with multiple angles
– Solve trig equations quadratic in form
– Use factoring to separate different functions
in trig equations
– Use identities to solve trig equations
– Use a calculator to solve trig equations
What is SOLVING a trig equation?
• It means finding the values of x that will
make the equation true. (Just as we did
with algebraic equations!)
• Until now, we have worked with identities,
equations that are true for ALL values of x.
Now we’ll be solving equations that are
true only for specific values of x.
Is this different that solving
algebraic equations?
• Not really, but sometimes we utilize trig
identities to facilitate solving the equation.
• Steps are similar: Get function in terms of
one trig function, isolate that function, then
determine what values of x would have
that specific value of the trig function.
• You may also have to factor, simplify, etc,
just as if it were an algebraic equation.
Solve:
4 cos x  6 cos x  2
2
2 cos x  3 cos x  1
2
2 cos x  3 cos x  1  0
factor : (2 cos x  1)(cos x  1)  0
(2 cos x  1  0)OR (cos x  1  0)
1
(2 cos x  1, cos x  )OR (cos x  1)
2
2
4
(x 
 2n,
 2n)OR ( x    2n)
3
3
2