14-6 Angle Identities
Quick Review
Angle identities are used to solve trigonometric equations.
Negative angle identities
sin
(
−
θ
)
=
−
sin
θ
sine open negative theta close equals negative sine theta
cos
(
−
θ
)
=
cos
θ
cosine open negative theta close equals cosine theta
tan
(
−
θ
)
=
−
tan
θ
tangent open negative theta close equals negative tangent theta
Cofunction identities
sin
(
π
2
−
θ
)
=
cos
θ
sine . open . pi over 2 , minus theta . close . equals cosine theta
cos
(
π
2
−
θ
)
=
sin
θ
cosine . open . pi over 2 , minus theta . close . equals sine theta
tan
(
π
2
−
θ
)
=
cot
θ
tangent . open . pi over 2 , minus theta . close . equals co-tangent theta
Angle difference identities
sin
(
A
−
B
)
=
sin
A
cos
B
−
cos
A
sin
B
sine open eh minus b close equals sine eh cosine b minus cosine eh sine b
cos
(
A
−
B
)
=
cos
A
cos
B
+
sin
A
sin
B
cosine open eh minus b close equals cosine eh cosine b plus sine eh sine b
tan
(
A
−
B
)
=
tan
A
−
tan
B
1
+
tan
A
tan
B
tangent . open , eh minus b , close . equals . fraction tangent eh minus tangent b , over 1 plus tangent eh tangent b end fraction
Angle sum identities
sin
(
A
+
B
)
=
sin
A
cos
B
+
cos
A
sin
B
sine open eh plus b close equals sine eh cosine b plus cosine eh sine b
cos
(
A
+
B
)
=
cos
A
cos
B
−
sin
A
sin
B
cosine open eh plus b close equals cosine eh cosine b minus sine eh sine b
tan
(
A
+
B
)
=
tan
A
+
tan
B
1
−
tan
A
tan
B
tangent . open , eh plus b , close . equals . fraction tangent eh plus tangent b , over 1 minus tangent eh tangent b end fraction
Example
What is the exact value of cos(165°)?
cos
165
°
=
cos
(
120
°
+
45
°
)
=
cos
120
°
cos
45
°
−
sin
120
°
sin
45
°
=
(
−
cos
60
°
)
cos
45
°
−
sin
60
°
sin
45
°
=
−
1
2
·
2
2
−
3
2
·
2
2
=
−
2
4
−
6
4
=
−
2
+
6
4
table with 6 rows and 2 columns , row1 column 1 , cosine , 165 degrees , column 2 equals cosine . open . 120 degrees plus 45 degrees . close , row2 column 1 , , column 2 equals cosine , 120 degrees cosine , 45 degrees negative sine , 120 degrees sine , 45 degrees , row3 column 1 , , column 2 equals . open . negative cosine , 60 degrees . close cosine , 45 degrees negative sine , 60 degrees sine , 45 degrees , row4 column 1 , , column 2 equals negative , 1 half , middle dot , fraction square root of 2 , over 2 end fraction , minus , fraction square root of 3 , over 2 end fraction , middle dot , fraction square root of 2 , over 2 end fraction , row5 column 1 , , column 2 equals negative , fraction square root of 2 , over 4 end fraction , minus , fraction square root of 6 , over 4 end fraction , row6 column 1 , , column 2 equals negative . fraction square root of 2 plus square root of 6 , over 4 end fraction , end table
Exercises
Verify each identity.
-
cos
(
θ
+
π
2
)
=
−
sin
θ
cosine . open . theta plus , pi over 2 . close . equals negative sine theta
-
sin
2
(
θ
−
π
2
)
=
cos
2
θ
sine squared . open . theta minus , pi over 2 . close . equals , cosine squared , theta
Find the exact value.
-
tan
15
°
tangent , 15 degrees
-
sin
300
°
sine , 300 degrees
-
cos
255
°
cosine , 255 degrees
- tan(
−
75
°
negative 75 degrees )
14-7 Double-Angle and Half-Angle Identities
Quick Review
You can use double-angle and half-angle identities to find exact values of trigonometric expressions. In the half-angle identities, choose the positive or negative sign for each function depending on the quadrant in which
A
2
eh over 2 lies.
Double-angle identities
|
cos
2
θ
=
cos
2
θ
−
sin
2
θ
cosine 2 theta equals , cosine squared , theta negative , sine squared , theta
|
cos
2
θ
=
2
cos
2
θ
−
1
cosine 2 theta equals 2 , cosine squared , theta negative 1
|
|
cos
2
θ
=
1
−
2
sin
2
θ
cosine 2 theta equals 1 minus 2 , sine squared , theta
|
sin
2
θ
=
2
sin
θ
cos
θ
sine 2 theta equals 2 sine theta cosine theta
|
|
tan
2
θ
=
2
tan
θ
1
−
tan
2
θ
tangent 2 theta equals . fraction 2 tangent theta , over 1 minus , tangent squared , theta end fraction
|
Half-angle identities
sin
A
2
=
±
1
−
cos
A
2
sine , eh over 2 , equals plus minus . square root of fraction 1 minus cosine eh , over 2 end fraction end root
cos
A
2
=
±
1
+
cos
A
2
cosine , eh over 2 , equals plus minus . square root of fraction 1 plus cosine eh , over 2 end fraction end root
tan
A
2
=
±
1
−
cos
A
1
+
cos
A
tangent , eh over 2 , equals plus minus . square root of fraction 1 minus cosine eh , over 1 plus cosine eh end fraction end root
Example
What is the exact value of cos 15°?
cos
75
°
=
cos
(
150
°
2
)
=
1
+
cos
150
°
2
=
1
−
3
2
2
=
2
−
3
4
=
2
−
3
2
table with 3 rows and 2 columns , row1 column 1 , cosine , 75 degrees , column 2 equals cosine . open . fraction 150 degrees , over 2 end fraction . close , row2 column 1 , , column 2 equals . square root of fraction 1 plus cosine , 150 degrees , over 2 end fraction end root , row3 column 1 , , column 2 equals . square root of fraction 1 minus , fraction square root of 3 , over 2 end fraction , over 2 end fraction end root . equals . square root of fraction 2 minus square root of 3 , over 4 end fraction end root . equals . fraction square root of 2 minus square root of 3 end root , over 2 end fraction , end table
Exercises
Use the double-angle identity to find the exact value of each expression.
-
sin
120
°
sine , 120 degrees
-
cos
30
°
cosine , 30 degrees
-
tan
300
°
tangent , 300 degrees
-
sin
240
°
sine , 240 degrees
