7.3 Sum and Difference Identities

Previous Identities

Reciprocal Identities

Quotient Identities

Pythagorean Identities

Negative-Number Identities

Note It will be necessary to recognize alternative forms of the identities

above, such as sin² = 1 – cos² and cos² = 1 – sin² .

sin1

csccos

1sec

tan1

cot

sincos

cotcossin

tan

222222 csccot1sec1tan1cossin

tan)tan(cos)cos(sin)sin(

Combined Sum and Difference Formulas

sinsincoscoscos

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sin α ± β( ) = sinα cosβ ± cosα sinβ

tantan1

tantantan

Example Find the exact value of the following.(a) cos 15°

(b) cos125

cos15 cos(45 30 )

cos45 cos30 sin 45 sin30

2 3 2 1 6 22 2 2 2 4

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cos5π12

= cos2π12

+3π12

⎛

⎝ ⎜

⎞

⎠ ⎟= cos

π6

+π4

⎛

⎝ ⎜

⎞

⎠ ⎟= cos(45°+30°)

= cosπ6

cosπ4

−sinπ6

sinπ4

=3

2⋅ 2

2−

12⋅ 2

2=

6− 24

(or 60° – 45°)

Example Find the exact value of the following.

(a) sin 75°(b) tan (c) sin 40° cos 160° – cos 40° sin 160°

Solution(a)

127

sin 75 sin(45 30 )

sin 45 cos30 cos 45 sin 30

2 3 2 1 6 2

2 2 2 2 4

(b)

(c) sin 40°cos 160° – cos 40°sin 160° =sin(40°-160°) = sin(–120°)

32

13113

4tan

3tan1

4tan

3tan

43tan

127

tan

23

Find the exact value of each expression:

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a) cosπ3

cosπ6

+sinπ3

sinπ6

b) sinπ3

cosπ6

− cosπ3

sinπ6

c)tanπ3

+ tanπ6

1− tanπ3

tanπ6

=0

=1

Undefined

.12

sin of eexact valu theFind

34sin

12sin

sin cos sin cos 4 3 3 4

2

2

2

3

2

1

2

2

24

64

2 64

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Find(tan5π

12) =

tan(3π

12+

2π

12) =

tan(π

4+π

6) =

tanπ

4+ tan

π

6

1− tanπ

4tanπ

6

=1+

3

3

1−1⋅3

3

=3+ 3

3− 3

EXAMPLE:

Find the exact value of ( cos 80° cos 20° + sin 80° sin 20°) .

Solution The given expression is the right side of the formula for cos( - ) with = 80° and = 20°.

cos 80° cos 20° + sin 80° sin 20° = cos (80° - 20°)

= cos 60° = 1/2

cos( -) = cos cos + sin sin

Example

Example

12sin

12

7cos

12cos

12

7sin

Write the following expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.

Solution:

12

sin

12

6sin

1212

7sin

12sin

12

7cos

12cos

12

7sin

Evaluating a Trigonometric Equation• Find the exact value of :

sin42 cos12 cos42 sin12

sin140 cos50 cos140 sin50

tan140 tan601 tan140 tan60

An Application of a Sum Formula• Write as an algebraic

expression

)arccos2sin(arctan

)arccos1cos(arctan

xx

x

Proving a Cofunction Identity

xxxx cos2

sin sin2

cos