Post on 27-Dec-2015
Trig – 04/19/23
Simplify.
312 Homework: p382 VC, 1-8, 17-25 odds
Honors: 27-30 all
Today’s Lesson: Double-Angle & Power-Reducing Formulas
€
sin3π
2− x
⎛
⎝ ⎜
⎞
⎠ ⎟
Trig/Pre-Calculus
You will:• Use the double-angle formulas to evaluate trig
functions.• Use the double-angle formulas to solve trig equations.
Honors• Use the Power-Reducing formulas to evaluate trig
functions.• Use the Power-Reducing formulas to solve trig
functions.
Today’s Lesson: Double-Angle & Power-Reducing Formulas
Double Angle Formulas
€
sin2u=2sinucosu
€
cos2u = cos2 u − sin2 u
= 2cos2 u −1
=1 − 2sin2 u
€
tan2u=2 tanu
1−tan2 u
Double Angle FormulasFind the exact value of each trig function.1. sin x 4. sin 2x
2. cos x 5. cos 2x
3. tan x 6. tan 2x
x 5
12
Double Angle FormulasFind the exact value of each trig function.1. sin x 4. sin 2x
2. cos x 5. cos 2x
3. tan x 6. tan 2x
x8
15
Double Angle FormulasUse the following to find and
–12
5
13
€
cosθ =513
, 3π2
<θ < 2π
€
sin2θ, cos2θ
€
tan2θ.
€
θ
€
sin2θ =2sinθcosθ
€
=2 −1213
⎛
⎝ ⎜
⎞
⎠ ⎟513
⎛
⎝ ⎜
⎞
⎠ ⎟
€
=−120169
€
cos2θ =2cos2θ −1
€
=2513
⎛
⎝ ⎜
⎞
⎠ ⎟2
−1
€
=225169
⎛
⎝ ⎜
⎞
⎠ ⎟−1
€
=50169
−169169
€
=−119169
Double Angle FormulasUse the following to find and
–12
5
13
€
cosθ =513
, 3π2
<θ < 2π
€
sin2θ, cos2θ
€
tan2θ.
€
θ
€
tan2θ =2 tanθ
1−tan2θ
€
=2 −12
5
⎛
⎝ ⎜
⎞
⎠ ⎟
1− −125
⎛
⎝ ⎜
⎞
⎠ ⎟2
€
=−245
2525
−14425
€
=−245
−11925
€
=120119
Double Angle FormulasUse the following to find and
4
35
€
sinθ =35, 0 <θ <
π2
€
sin2θ, cos2θ
€
tan2θ.
€
θ
€
sin2θ =2sinθcosθ
€
=235
⎛
⎝ ⎜
⎞
⎠ ⎟45
⎛
⎝ ⎜
⎞
⎠ ⎟
€
=2425
€
cos2θ =2cos2θ −1
€
=245
⎛
⎝ ⎜
⎞
⎠ ⎟2
−1
€
=21625
⎛
⎝ ⎜
⎞
⎠ ⎟−1
€
=3225
−2525
€
=725
Double Angle FormulasUse the following to find and
€
sinθ =35, 0 <θ <
π2
€
sin2θ, cos2θ
€
tan2θ.
€
tan2θ =2 tanθ
1−tan2θ
€
=2 34
⎛
⎝ ⎜
⎞
⎠ ⎟
1−34
⎛
⎝ ⎜
⎞
⎠ ⎟2
€
=
32
1616
− 916
€
=
32716
€
=247
4
35
€
θ
Solving a Multiple-Angle EqSolve.
€
2cos x + sin2x = 0
€
sin 2x = 2sin x cosx
€
2cos x + 2sin x cos x = 0
€
2cos x 1+ sin x( ) = 0
€
2cos x = 0
€
1+ sin x = 0
€
cos x = 0
€
x =π
2 ,
3π
2€
sin x = −1
€
+2nπ + 2nπ
Double-Angle Formula
€
=π2
+ nπ
Solving a Multiple-Angle EqSolve.
€
cos2x + cos x = 0
€
cos2x = 2cos2 x −1
€
2cos2 x −1+ cos x = 0
€
2cos x −1( ) cos x +1( ) = 0
€
x =π
3 ,
5π
3
€
+2nπ + 2nπ
Double-Angle Formula
€
2cos x −1 = 0
€
cos x +1 = 0
€
cos x =1
2
€
cos x = −1
€
2cos2 x+ cosx−1=0
€
x = π
€
+2nπ
Power-Reducing FormulasHonors
€
sin2 u=1−cos2u
2
€
cos2 u=1+ cos2u
2
€
tan2 u=1−cos2u1+ cos2u
Power Reducing FormulasRewrite sin2x as a sum of first powers of cosines of multiple angles.
2
2cos1sin2 x−
=
Power-Reducing FormulasRewrite sin4x as a sum of first powers of cosines of multiple angles.
€
sin4 = sin2 x( )2
2
2
2cos1⎟⎠⎞
⎜⎝⎛ −
=x
€
=1 − cos2x( )
2
22
€
=1
41 − cos2x( )
2
( )xx 2cos2cos214
1 2+−=
€
=1
41 − 2cos2x + cos2 2x( )
( )⎟⎠⎞
⎜⎝⎛ +
+−=2
22cos12cos21
4
1 xx
xx 4cos8
1
8
12cos
2
1
4
1++−=
xx 4cos8
12cos
2
1
8
3+−=
( )xx 4cos2cos438
1+−=
Power Reducing FormulasRewrite tan4x as a sum of first powers of cosines of multiple angles.