7.3 sum and difference identities. previous identities reciprocal identities quotient identities...
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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
€
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
€
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:
€
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
€
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
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