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7.3 Sum and Difference Identities
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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(
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Combined Sum and Difference Formulas
sinsincoscoscos
€
sin α ± β( ) = sinα cosβ ± cosα sinβ
tantan1
tantantan
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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°)
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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
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(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
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Find the exact value of each expression:
€
a) cosπ3
cosπ6
+sinπ3
sinπ6
b) sinπ3
cosπ6
− cosπ3
sinπ6
c)tanπ3
+ tanπ6
1− tanπ3
tanπ6
=0
=1
Undefined
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.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:
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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
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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
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Evaluating a Trigonometric Equation• Find the exact value of :
sin42 cos12 cos42 sin12
sin140 cos50 cos140 sin50
tan140 tan601 tan140 tan60
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An Application of a Sum Formula• Write as an algebraic
expression
)arccos2sin(arctan
)arccos1cos(arctan
xx
x
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Proving a Cofunction Identity
xxxx cos2
sin sin2
cos