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
xx
xx
xx
22
22
22
csc1cot
sec1tan
1cossin
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:
• Manipulate right to look like left. Expand the binomial and express in terms of sin & cos
x
x
xx
xx
x
x
x
x
x
xx
x
x
x
x
xx
xxxxxx
cos1
cos1
)cos1)(cos1(
)cos1)(cos1(
cos1
)cos1(
sin
)cos1(
sin
coscot21
sin
cos
sin
cot
sin
2
sin
1
cotcotcsc2csc)cot(csc
2
2
2
2
2
2
2
2
2
222
x
xxx
cos1
cos1)cot(csc 2
5.2 Sum & DifferenceFormulas
• 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 + sinAsinBcos(A+B) = cosAcosB - sinAsinB
• Use difference formula to find cos(165 degrees)
4
622
2
6
4
2
2
23
2
2
2
1
45sin210sin45cos210cos
)45210cos()165cos(
sin(A+B) = sinAcosB + cosAsinBsin(A-B) = sinAcosB - cosAsinB
4
62
2
3
2
2
2
1
2
2
3sin
4
3cos
3cos
4
3sin
34
3sin
12
4
12
9sin
12
13sin
)12
13(sin
find
33
33
33
11
33
1
6tan
4tan1
6tan
4tan
)64
tan(
)12
2
12
3tan()
12
5(tan
tantan1
tantan)tan(
tantan1
tantan)tan(
Find
BA
BABA
BA
BABA
5.3 Double-Angle, Power-Reducing, & 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)
A
AA
AAA
AAA
2
22
tan1
tan22tan
sincos2cos
cossin22sin
You use these identities to find exact values of trig functions of “non-
special” angles and to verify other identities.
Double-angle formula for cosine can be expressed in other ways
1cos2
)cos1(cossincos2cos
sin21
sin)sin1(sincos2cos
2
2222
2
2222
A
AAAAA
A
AAAAA
We can now develop the Power-Reducing Formulas.
A
AAthus
AA
AA
AAcall
AA
AA
AAcall
2cos1
2cos1tan:
2
2cos1sin
2cos1sin2
sin212cos:Re
cos2
12cos
cos212cos
,1cos22cos:Re
2
2
2
2
2
2
2
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.
2
cos1
2sin
2sin
2
cos1
22
2cos1
2sin
22:Re
2
2
xxx
xx
x
xx
call
Half-angle identities for tangent
A
AAA
AA
cos1
sin
2tan
sin
cos1
2tan
5.4 Product-to-Sum & Sum-to-Product Formulas
• Objectives
–Use the product-to-sum formulas
–Use the sum-to-product formulas
Product to Sum Formulas
)]sin()[sin(2
1sincos
)]sin()[sin(2
1cossin
)]cos()[cos(2
1coscos
)]cos()[cos(2
1sinsin
BABABA
BABABA
BABABA
BABABA
Sum-to-Product Formulas
2sin
2sin2coscos
2cos
2cos2coscos
2cos
2sin2sinsin
2cos
2sin2sinsin
BABABA
BABABA
BABABA
BABABA
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:
)2()23
4,2
3
2(
)1(cos)2
1cos,1cos2(
)01(cos)01cos2(
0)1)(cos1cos2(:
01cos3cos2
1cos3cos2
2cos6cos4
2
2
2
nxORnnx
xORxx
xORx
xxfactor
xx
xx
xx