slide 6- 1 copyright © 2006 pearson education, inc. publishing as pearson addison-wesley

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Trigonometric Identities, Inverse Functions,

and Equations

Chapter 6


6.1Identities: Pythagorean and

Sum and Difference State the Pythagorean identities.

Simplify and manipulate expressions containing trigonometric expressions.

Use the sum and difference identities to find function values.


Basic Identities

An identity is an equation that is true for all possible replacements of the variables.

1 1sin , csc ,

csc sin1 1

cos , sec ,sec cos

1 1tan , cot ,

cot tan

x xx x

x xx x

x xx x

sin( ) sin ,

cos( ) cos ,

tan( ) tan ,

sintan ,

coscos

cotsin

x x

x x

x x

xx

xx

xx


Pythagorean Identities

2 2

2 2

2 2

sin cos 1,

1 cot csc ,

1 tan sec

x x

x x

x x


Example

Multiply and simplify:

a)

Solution:

sin (cot csc )x x x

sin (cot csc )

sin cot sin csc

cos 1sin sin

sin sincos 1

x x x

x x x x

xx x

x xx


Example continued

b) Factor and simplify:

Solution:

4 2 2sin sin cosx x x

4 2 2

2 2 2

2

2

sin sin cos

sin (sin cos )

sin (1)

sin

x x x

x x x

x

x


Another Example

Simplify the following

trigonometric expression:

Solution:

cos cos

1 sin 1 sin

x x

x x

2 2

2

2

cos (1 sin ) cos (1 sin )

(1 sin )(1 sin ) (1 sin )(1 sin )

cos sin cos cos sin cos

1 sin 1 sin2cos

1 sin2cos

cos2

or 2seccos

x x x x

x x x x

x x x x x x

x xx

xx

x

xx


Sum and Difference Identities

There are six identities here, half of them obtained by using the signs shown in color.

sin( ) sin cos cos sin ,

cos( ) cos cos sin sin ,

tan tantan( )

1 tan tan

u v u v u v

u v u v u v

u vu v

u v


Example

Find sin 75 exactly.

sin 75 sin(30 45 )

sin30 cos45 cos30 sin 45

1 2 3 2

2 2 2 2

2 6

4 4

2 6

4


6.2 Identities: Cofunction,

Double-Angle, and Half-Angle Use cofunction identities to derive other identities.

Use the double-angle identities to find function values of twice an angle when one function value is known for that angle.

Use the half-angle identities to find function values of half an angle when one function value is known for that angle.

Simplify trigonometric expressions using the double-angle and half-angle identities.


Cofunction Identities

Cofunction Identities for the Sine and Cosine

sin cos , cos sin ,2 2

tan cot , cot tan ,2 2

sec csc , csc sec2 2

x x x x

x x x x

x x x x

sin cos cos sin2 2

x x x x


Example

Find an identity for

Solution:

cot .2

x

cos2

cot2 sin

2

sin

costan

xx

x

x

xx


Double-Angle Identities

2

sin 2 2sin cos ,

2 tantan 2

1 tan

x x x

xx

x

2 2

2

2

cos2 cos sin

1 2sin

2cos 1

x x x

x

x


Example

Find an equivalent expression for cos 3x.

Solution:

2

2 2

2

cos3 cos(2 )

cos2 cos sin 2 sin

(1 2sin )cos 2sin cos sin

cos 2sin cos 2sin cos

cos 4sin cos

x x x

x x x x

x x x x x

x x x x x

x x x


Half-Angle Identities

1 cossin ,

2 2

1 coscos ,

2 2

1 costan

2 1 cossin 1 cos

1 cos sin

x x

x x

x x

xx x

x x


Example

Find sin ( /8) exactly.

Solution: 1 cos

4 4sin2 2

21

22

2 2

2


Another Example

Simplify .

Solution:

tan tan 12

xx

1 cos sintan tan 1 1

2 sin cossin (1 cos )

1sin cos

1 cos1

cos1 cos

1cos cossec 1 1

sec

x x xx

x xx x

x xx

xx

x xx

x


6.3Proving

Trigonometric Identities Prove identities using other identities.


The Logic of Proving Identities

Method 1: Start with either the left or the right side of the equation and obtain the other side.

Method 2: Work with each side separately until you obtain the same expression.


Hints for Proving Identities

Use method 1 or 2. Work with the more complex side first. Carry out any algebraic manipulations, such as adding,

subtracting, multiplying, or factoring. Multiplying by 1 can be helpful when rational

expressions are involved. Converting all expressions to sines and cosines is often

helpful. Try something! Put your pencil to work and get

involved. You will be amazed at how often this leads to success.


Example

Prove the identity .

Solution: Start with the left side.

2 2 2(tan 1)(cos 1) tanx x x

2 2 2

22 2

2

22 2 2

2

22 2 2

2

(tan 1)(cos 1) tan

sin1 (cos 1) tan

cos

sinsin cos 1 tan

cos

sinsin cos 1 tan

cos

x x x

xx x

x

xx x x

x

xx x x

x

22

2

22

2

2 2

sin1 1 tan

cos

sintan

cos

tan tan

xx

x

xx

x

x x


Another Example

Prove the identity:

Solution: Start with the right side.

Solution continued1

csc sinsec tan

x xx x

2

1csc sin

sec tan1

sinsin

1 sin

sin sin

x xx x

xx

x

x x

2

2

1 1 sin

sec tan sin

cos

sincos cos

sin 1cot cos

1 1

tan sec1 1

sec tan sec tan

x

x x x

x

xx x

xx x

x x

x x x x


Example

Prove the identity .

Solution: Start with the left side.

tan cottan cot

tan cot

x yy x

x y

tan cottan cot

tan cot

tan cot

tan cot tan cot

1 1

cot tan

tan cot tan cot

x yy x

x y

x y

x y x y

y x

y x y x


6.4Inverses of the Trigonometric

Functions Find values of the inverse trigonometric functions.

Simplify expressions such as sin (sin–1 x) and sin–1 (sin x).

Simplify expressions involving compositions such as sin (cos–1 ) without using a calculator.

Simplify expressions such as sin arctan (a/b) by making a drawing and reading off appropriate ratios.

1

2


Inverse Trigonometric Functions

[0, ][1, 1]

[1, 1]

RangeDomainFunction

1sin

arcsin , where sin

y x

x x y

1cos

arccos , where cos

y x

x x y

1tan

arctan , where tan

y x

x x y

( , )

[ / 2, / 2]

/ 2, / 2


Example

Find each of the following:

a)

b)

c)

Solution: a) Find such that

would represent a 60° or 120° angle.

1 3sin

2

1 3cos

2

1tan ( 1)

3sin

2

1 3sin 60 and 120

22

or and 3 3


Solution continued

b) Find such that

would represent a 30° reference angle in the 2nd and 3rd quadrants.

Therefore, = 150° or 210°

c) Find such that

This means that the sine and cosine of must be opposites.

Therefore, must be 135° and 315°.

3cos

2

1 3cos 150 and 210

2

5 7or and

6 6

tan 1.

1tan ( 1) 135 and 315

3 7or and

4 4


Domains and Ranges


Composition of Trigonometric Functions

1 1

1 1

1 1

sin(sin ) , for all in the domain of sin .

cos(cos ) , for all in the domain of cos .

tan(tan ) , for all in the domain of tan .

x x x

x x x

x x x


Examples

Simplify:

Since 1/2 is in the domain of sin–1,

Simplify:

Since is not in the

domain of cos–1,

1 1sin sin

2

1 1 1sin sin

2 2

1 3 2cos cos

2

3 2

2

1 3 2cos cos

2

does not exist.


Special Cases

1 1

1 1

1 1

sin (sin ) , for all in the range of sin .

cos (cos ) , for all in the range of cos .

tan (tan ) , for all in the range of tan .

x x x

x x x

x x x


Examples

Simplify:

Since /2 is in the range of sin–1,

Simplify:

Since /3 is in the range of tan–1,

1sin sin2

1tan tan3

1sin sin2 2

1tan tan3 3


More Examples

Simplify:

Solution:

Simplify:

Solution:

1 1sin cos

2

1 1 2cos 120 or

2 3

2 3sin

3 2

1 2tan sin

3

1

2 3sin

3 2

3tan 40.9

2


6.5 Solving

Trigonometric Equations Solve trigonometric equations.


Solving Trigonometric Equations

Trigonometric Equation—an equation that contains a trigonometric expression with a variable.

To solve a trigonometric equation, find all values of the variable that make the equation true.


Example

Solve 2 sin x 1 = 0. Solution: First, solve for

sin x on the unit circle.

The values /6 and 5/6 plus any multiple of 2 will satisfy the equation. Thus the solutions are

where k is any integer.1

2sin 1 0

2sin 1

1sin

21

sin2

530 ,150 or ,

6 6

x

x

x

x

x

52 and 2

6 6k k


Graphical Solution

We can use either the Intersect method or the Zero method to solve trigonometric equations. We graph the equations y1 = 2 sin x 1 and y2 = 0.


Another Example

Solve 2 cos2 x 1 = 0.

Solution: First, solve for cos x on the unit circle.2

2

2

2cos 1 0

2cos 1

1cos

2

1cos

2

2cos

2

x

x

x

x

x

45 ,135 ,225 ,315

3 5 7 , , ,

4 4 4 4

x

or

3 5 7The values , , , plus

4 4 4 4any multiple of 2 will satisfy

the equation.

The solution can be written

as where is any integer.4 2

k k


Graphical Solution

Solve 2 cos2 x 1 = 0. One graphical solution shown.


One More Example

Solve 2 cos x + sec x = 0

Solution:

2

2cos sec 0

12cos 0

cos1

2cos 1 0cos

x x

xx

xx

2

2

2

2cos 1 0

2cos 1

1cos

2

1cos

2

x

x

x

x

1cos

2x

Since neither factor of the equation can equal zero, the equation has no solution.

10 or

cos1

0cos

x

x


Graphical Solution

2 cos x + sec x


Last Example

Solve 2 sin2 x + 3sin x + 1 = 0. Solution: First solve for sin x on the unit circle.

22sin 3sin 1 0

(2sin 1)(sin 1) 0

x x

x x

1

2sin 1 0

2sin 1

1sin

21

sin2

7 11,

6 6

x

x

x

x

x


Last Example continued

where k is any integer.

One Graphical Solution

1

sin 1 0

sin 1

sin ( 1)

3

2

x

x

x

x

7 11 32 , 2 , 2

6 6 2x k k k

slide 6- 1 copyright © 2006 pearson education, inc. publishing as pearson addison-wesley

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