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.