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