Double-Angle and Half-Angle Formulas

Double-Angle and Half-Angle Formulas

Section 5.5

In the previous sections, we used:

a) The Fundamental Identitiesa) Sin²x + Cos²x = 1

b) Sum & Difference Formulasa) Cos (u – v) = Cos u Cos v + Sin u Sin v

Now we will use double angle and half angle formulas

Multiple-Angle Formulas

Double-angle formulas are the formulas used most often:

Double-Angle Formulas

u Cosu Sin 2 2u Sin uSin -u Cos 2u Cos 221 -u 2Cos 2

u2Sin - 1 2uTan - 1u2Tan 2u Tan 2

Use the following triangle to find the following:


2

5

Sin 2θ

Cos 2θ

Tan 2θ

29

θ



2

5

Sin 2θ29

= 2Sin θ Cos θ

2

292

θ

295

2920


2

5

Cos 2θ

29

= 2Cos² θ - 1

2

θ

2

295

129252

1-

12950

2921


2

5

Tan 2θ

29

2Tan - 1

2Tan

θ 2

521

522

2521

54

254 -15

4

2120



1

4

Csc 2θ

Sec 2θ

Cot 2θ

17

θ

817

1517

815

General guidelines to follow when the double-angle formulas to solve equations:

1) Apply the appropriate double-angle formula

2) Look to factor

3) Solve the equation using the different strategies involved in solving equations


Solve the following equation in the interval [0, 2π)


Sin 2x – Cos x = 01. Apply the double-angle formula

2 Sin x Cos x – Cos x = 02. Look to factor

Cos x (2 Sin x – 1) = 0

Double-Angle FormulasCos x (2 Sin x – 1) = 0

3. Solve the equationCos x = 0 2 Sin x - 1= 0

Sin x = ½ ,

2

2

3x

x ,6

6

5

Solve the following equation in the interval [0, 2π)


2 Cos x + Sin 2x = 02 Cos x + 2 Sin x Cos x = 02 Cos x (1+ Sin x) = 0

2 Cos x = 0 1 + Sin x = 0

Double-Angle Formulas2 Cos x = 0 1 + Sin x = 0 Cos x = 0 Sin x = -1

,2

2

3x x

23

Solve the following equations for x in the interval [0, 2π)

a) Sin 2x Sin x = Cos x

b) Cos 2x + Sin x = 0


,2

x ,2

3 ,6 ,

65 ,

67

611

,2

x ,6

76

11

Double-Angle FormulasSin 2x Sin x = Cos x2 Sin x Cos x Sin x = Cos x2 Sin²x Cos x – Cos x = 0Cos x (2 Sin²x – 1) = 0Cos x = 0 2 Sin²x – 1 = 0

Sin²x = ½ Sin x = ± ½

x =

x ,2

,2

3

,6 ,

65 ,

67

611

Double-Angle FormulasCos 2x + Sin x = 01 – 2Sin² x + Sin x = 02Sin² x - Sin x - 1= 0

(2 Sin x + 1) (Sin x – 1) = 02 Sin x + 1 = 0 Sin x – 1 = 0 Sin x = ½ Sin x = 1

xx = ,2

,6

76

11


Section 5.5

Evaluating Functions Involving Double Angles

Use the given information to find the following:

Sin 2x Cos 2x Tan 2x


1312 Sin x x

2


1312 Sin x x

2 12 13

x-5Sin 2x = 2Sin x Cos x

2

1312

135

169120-


1312 Sin x x

2 12 13

x-5Cos 2x = 2Cos² x - 1

2 2

135

1

1169252

1

16950

169119-


12 13x

-5Tan 2x

2

5-12-1

5-122

xTan - 12Tan x 2

25144 -1

5-24

25

119- 5-

24

119 120

Evaluating Functions Involving Double Angles

Use the given information to find the following:

Sin 2x Cos 2x Tan 2x


178 x Cos 2 x

23


178 x Cos 2 x

23

-1517

x8

Sin 2x = 2Sin x Cos x2

1715-

178

289240-


Cos 2x = 2Cos² x - 12

2

178

1

1289642

1

289128

289161-

158 x Cos 2 x

23

-1517

x8


2

815--1

815-2

64

225 -18

30-

64161-

830-

161 240

-1517

x8

Tan 2x xTan - 1

2Tan x 2

The next (and final) set of formulas we have are called half-angle formulas.

2uSin

2u Cos

2uTan

2u Cos1

2u Cos1

uSin u Cos - 1

u Cos 1uSin

The sign of Sin and Cos depend on what quadrant u/2 is in

Use the following triangle to find the six trig functions of θ/2

Half-Angle Formulas

257θ

2

Sin

2

Cos

2

Tan

102

7110

27

Half-Angle Formulas257

θ

2Sin

242

u Cos1

225

241

225

1

501

501

25

1

102


θ

2 Cos

242

u Cos1

225

241

225

49

5049

507

25

7

1027


θ

2Tan

24

257

25241

25

725

1

71

uSin u Cos - 1

Find the exact value of the Cos 165º.

Half-Angle Formulas

165º is half of what angle?

Cos 165º = 2

330 Cos

2

330 Cos2

330 Cos1

22

31

22

32

432

2

32

Find the exact value of the Sin 105º.

Half-Angle Formulas


Sin 105º = 2

210Sin

2

210Sin 2

210 Cos12

231

22

32

432

2

32

Find the exact value of the Tan 15º.

Half-Angle Formulas


Tan 15º = 230Tan

2

30Tan 30Sin

30 Cos1

21

23 - 1

2

12

32

32


Section 5.5

Half-Angle Formulas:Find , x

2 and

1312 Sin x Given

2xTan c)

2x Cos b)

2xSin a)

1312x

-5

13133

13132

23

Half-Angle Formulas

2xSin a) 1312

x-5

2u Cos1

213

5-1

213

51

213

18

2618

139

133

13

133

Half-Angle Formulas

2x Cos b) 1312

x-5

a2

u Cos1

213

5-1

213

8

268

134

132

13

132

Half-Angle Formulas

2xTan c) 1312

x-5

Sin x xCos 1

1312

135- - 1

13

1213

5 1

1312

1318

1218

23

Half-Angle Formulas:Find ,

2 x 0 and

43 Tan x Given

2xCot c)

2x Sec b)

2x Csc a)

4

3x

510

310

3

Half-Angle Formulas

2xSin a)

2u Cos1

25

41

25

1 10

1

101

10

4

3x

5

Half-Angle Formulas

2x Cos b) a

2u Cos1

25

41

25

9

109

103

310

4

3x

5

Half-Angle Formulas

2xTan c)

Sin x xCos 1

53

54 - 1

5

35

1

31

34

3x

5

Solving Equations using the half-angle formulas:

1) Apply the appropriate formula2) Use the various methods we have learned

to solve equations1) Factor2) Combine Like Terms3) Isolate the Trig Function4) Solve the Equation for an Angle(s)

Half-Angle Formulas

Solve the following equation for x in the interval [0, 2π)

Half-Angle Formulas

xCos 2x Sin 2 2

xCos

2 xCos - 1 2

2

xCos 2

xCos - 1 2

xCos x Cos - 1

x2Cos 1 21 x Cos x ,

3

35


Half-Angle Formulas

2x 2Cos x Sin - 2 22

2

xCos 12 x Sin - 2 2

2

2

xCos 12 xSin - 2 2

Half-Angle Formulas

2

xCos 12 xSin - 2 2

xCos 1 xSin - 2 2 xCos 1 x)Cos - (1 - 2 2

xCos 1 xCos 1 - 2 2 0 xCos -x Cos 2

0 1) - x (Cos x Cos 1 x Cos 0 x Cos

,2

x ,

23 0


Half-Angle Formulas

0 1 - x Cos 2xSin

0 1 - x Cos 2

xCos 1

x Cos - 1 2

xCos 1

) ( 2

2

Half-Angle Formulas x Cos - 1

2 xCos 1

) ( 22

2

xCos1

xCos x 2Cos - 1 2

xCos2 x 4Cos - 2 x Cos - 1 2

0 1 x 3Cos -x Cos2 2 0 1) - x (Cos 1) - x Cos2(

0 1) - x (Cos 1) - x Cos2(

0 1 - x Cos 0 1 - x Cos2

21 xCos 1 xCos

,3

x

35

0 x

Because we squared both sides, check your answers!

Double-Angle and Half-Angle Formulas

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