Download Notes 17 (5.2, 5.3) Sum, Difference, Double, and Half

Notes 17a
Name ________________________
Sum and Difference Formulas (5.2)
Sum and Difference Formulas
1. cos (α=
+ β ) cos α cos β − sin α sin β
2. cos (α=
− β ) cos α cos β + sin α sin β
3. sin (α=
+ β ) sin α cos β + cos α sin β
4. sin (α=
− β ) sin α cos β − cos α sin β
tan α + tan β
5. tan (α + β ) =
1 − tan α tan β
tan α − tan β
6. tan (α − β ) =
1 + tan α tan β
Using the Sum or Difference Formulas (Forward): Use the sum or difference formulas to evaluate.
 π 2π 
1) cos (135 − 60 )
2) tan  +

4 3 
Using the Sum or Difference Formulas (Backwards): Find the exact value of each expression.
π
5π
− tan
tan
18
36
3) cos 70 cos 40 + sin 70 sin 40
4)
π
5π
1 + tan
tan
18
36
Finding the Exact Value: Find the exact value of each trigonometric function.
5π
sin
6) tan 75
5)
12
Using Sum and Difference:
4
1
Suppose that sin α = for a quadrant I angle α and cos β = for a quadrant I angle β .
2
5
7) cos (α + β )
8) tan(α + β )
Verifying Identities Using the Sum or Difference Formulas: Verify the following identities.
cos (α − β )
π
 cos θ − sin θ
10) tan  − θ  =
9)
= 1 + tan α tan β
cos α cos β
4
 cos θ + sin θ
Notes 17b
Name ________________________
Double-Angle and Half-Angle Formulas (5.3)
Double-Angle Formulas
1. sin 2θ = 2sin θ cos θ
2. cos
=
2θ cos 2 θ − sin 2 θ = 2 cos 2 θ − 1
2 tan θ
3. tan 2θ =
1 − tan 2 θ
= 1 − 2sin 2 θ
Using the Double-Angle Formulas (Forward): Use the given information to find the exact value.
12
1) Find sin 2θ if sin θ = , θ lies in quadrant II.
13
2)
Find tan 2θ if cot θ = 3, θ lies in quadrant III.
Using the Double-Anlge Formulas (Backwards): Find the exact value of each expression.
2 tan 22.5
π
3) 1 − 2sin 2
4)
1 − tan 2 22.5
12
Verifying Identities Using the Sum or Difference Formulas: Verify the following identities.
cos 2 x
2 cot θ
6) sin 2θ =
5)
= 1 − tan 2 x
2
1 + cot 2 θ
cos x
Half-Angle Formulas
x
1 − cos x
= ±
2
2
1 + cos x
x
2. cos = ±
2
2
x
1 − cos x
3. tan = ±
2
1 + cos x
1. sin
=
1 − cos x
sin x
=
sin x
1 + cos x
Using the Half-Angle Formulas (Forward): Use the given information to find the exact value.
8
θ
7) Find cos if tan θ = − , θ lies in quadrant II.
2
15
8)
Find tan
θ
2
if sec θ = −3, θ lies in quadrant III.
Find the Value: Use a half-angle formula to find the exact value of each expression.
3π
9) sin 22.5
10) tan
8