Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1...

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Double-Angle and Half-Angle Formulas Section 5.5

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Transcript of Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1...

Page 1: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle and Half-Angle Formulas

Section 5.5

Page 2: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

Page 3: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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 - 1

u2Tan 2u Tan

2

Page 4: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Use the following triangle to find the following:

Double-Angle Formulas

2

5

Sin 2θ

Cos 2θ

Tan 2θ

29

θ

Page 5: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Use the following triangle to find the following:

Double-Angle Formulas

2

5

Sin 2θ29

= 2Sin θ Cos θ

2

29

2θ

29

5

29

20

Page 6: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

2

5

Cos 2θ

29

= 2Cos² θ - 1

2

θ

2

29

5

129

252

1-

129

50

29

21

Page 7: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

2

5

Tan 2θ

29

2Tan - 1

2Tan

θ 2

52

1

52

2

2521

54

254 -15

4

21

20

Page 8: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Use the following triangle to find the following:

Double-Angle Formulas

1

4

Csc 2θ

Sec 2θ

Cot 2θ

17

θ

8

17

15

17

8

15

Page 9: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

Double-Angle Formulas

Page 10: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

Double-Angle Formulas

Sin 2x – Cos x = 0

1. Apply the double-angle formula2 Sin x Cos x – Cos x = 0

2. Look to factorCos x (2 Sin x – 1) = 0

Page 11: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

Cos x (2 Sin x – 1) = 0

3. Solve the equation

Cos x = 0 2 Sin x - 1= 0

Sin x = ½ ,

2

2

3x

x ,6

6

5

Page 12: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

Double-Angle Formulas

2 Cos x + Sin 2x = 0

2 Cos x + 2 Sin x Cos x = 0

2 Cos x (1+ Sin x) = 0

2 Cos x = 0 1 + Sin x = 0

Page 13: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

2 Cos x = 0 1 + Sin x = 0

Cos x = 0 Sin x = -1

,2

2

3x x

2

3

Page 14: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

a) Sin 2x Sin x = Cos x

b) Cos 2x + Sin x = 0

Double-Angle Formulas

,2

x ,

2

3,

6

,

6

5,

6

76

11

,2

x ,

6

76

11

Page 15: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

Sin 2x Sin x = Cos x2 Sin x Cos x Sin x = Cos x

2 Sin²x Cos x – Cos x = 0

Cos x (2 Sin²x – 1) = 0

Cos x = 0 2 Sin²x – 1 = 0Sin²x = ½ Sin x = ± ½

x =

x ,2

,

2

3

,6

,

6

5,

6

76

11

Page 16: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

Cos 2x + Sin x = 01 – 2Sin² x + Sin x = 0

2Sin² x - Sin x - 1= 0(2 Sin x + 1) (Sin x – 1) = 0

2 Sin x + 1 = 0 Sin x – 1 = 0 Sin x = ½ Sin x = 1

xx = ,2

,

6

76

11

Page 17: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle and Half-Angle Formulas

Section 5.5

Page 18: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Evaluating Functions Involving Double Angles

Use the given information to find the following:

Sin 2x Cos 2x Tan 2x

Double-Angle Formulas

13

12 Sin x

x 2

Page 19: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

13

12 Sin x

x 2

12 13

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

2

13

12

13

5

169

120-

Page 20: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

13

12 Sin x

x 2

12 13

x-5Cos 2x = 2Cos² x - 1

2 2

13

5

1

1169

252

1

169

50

169

119-

Page 21: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

12 13

x-5

Tan 2x

2

5-12

-1

5-12

2

xTan - 1

2Tan x

2

25144 -1

5-24

25

119- 5-

24

119

120

Page 22: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Evaluating Functions Involving Double Angles

Use the given information to find the following:

Sin 2x Cos 2x Tan 2x

Double-Angle Formulas

17

8 x Cos

2 x 2

3

Page 23: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

17

8 x Cos

2 x 2

3 -15

17

x8

Sin 2x = 2Sin x Cos x

2

17

15-

17

8

289

240-

Page 24: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

Cos 2x = 2Cos² x - 1

2 2

17

8

1

1289

642

1

289

128

289

161-

15

8 x Cos

2 x 2

3 -15

17

x8

Page 25: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle Formulas

2

815-

-1

815-

2

64

225 -18

30-

64161-

830-

161

240

-1517

x8

Tan 2x xTan - 1

2Tan x

2

Page 26: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

2

uSin

2

u Cos

2

uTan

2

u Cos1

2

u Cos1

uSin

u Cos - 1

u Cos 1

uSin

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

Page 27: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

Half-Angle Formulas

257

θ

2

Sin

2

Cos

2

Tan

10

2

7

1

10

27

Page 28: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas

257

θ

2

Sin

24

2

u Cos1

225

241

225

1

50

1

50

1

25

1

10

2

Page 29: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas

257

θ

2

Cos

24

2

u Cos1

225

241

225

49

50

49

50

7

25

7

10

27

Page 30: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas

257

θ

2

Tan

24

257

25241

25

725

1

7

1

uSin

u Cos - 1

Page 31: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Find the exact value of the Cos 165º.

Half-Angle Formulas

165º is half of what angle?

Cos 165º = 2

330 Cos

2

330 Cos

2

330 Cos1

22

31

22

32

4

32

2

32

Page 32: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Find the exact value of the Sin 105º.

Half-Angle Formulas

105º is half of what angle?

Sin 105º = 2

210Sin

2

210Sin

2

210 Cos12

231

22

32

4

32

2

32

Page 33: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Find the exact value of the Tan 15º.

Half-Angle Formulas

15º is half of what angle?

Tan 15º = 2

30Tan

2

30Tan

30Sin

30 Cos1

21

23 - 1

2

12

32

32

Page 34: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Double-Angle and Half-Angle Formulas

Section 5.5

Page 35: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas:Find , x

2 and

13

12 Sin x Given

2

xTan c)

2

x Cos b)

2

xSin a)

1312

x-5

13

133

13

132

2

3

Page 36: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas

2

xSin a) 1312

x-5

2

u Cos1

213

5-1

213

51

213

18

26

18

13

9

13

3

13

133

Page 37: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas

2

x Cos b) 1312

x-5

a2

u Cos1

213

5-1

213

8

26

8

13

4

13

2

13

132

Page 38: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas

2

xTan c) 1312

x-5

Sin x

xCos 1

1312

135- - 1

13

1213

5 1

1312

1318

12

18

2

3

Page 39: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas:Find ,

2 x 0 and

4

3 Tan x Given

2

xCot c)

2

x Sec b)

2

x Csc a)

4

3

x

510

3

10

3

Page 40: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas

2

xSin a)

2

u Cos1

25

41

25

1

10

1

10

1 10

4

3

x

5

Page 41: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas

2

x Cos b) a

2

u Cos1

25

41

25

9

10

9

10

3

3

10

4

3

x

5

Page 42: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas

2

xTan c)

Sin x

xCos 1

53

54 - 1

5

35

1

3

1 3

4

3

x

5

Page 43: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

Page 44: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

Half-Angle Formulas

xCos 2

x 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

3

5

Page 45: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

Half-Angle Formulas

2

x 2Cos x Sin - 2 22

2

xCos 12 x Sin - 2

2

2

2

xCos 12 xSin - 2 2

Page 46: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

,2

3 0

Page 47: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

Half-Angle Formulas

0 1 - x Cos 2

xSin

0 1 - x Cos 2

xCos 1

x Cos - 1 2

xCos 1

) (

2

2

Page 48: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

Half-Angle Formulas

x Cos - 1 2

xCos 1

) (

2

2

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(

Page 49: Section 5.5. In the previous sections, we used: a) The Fundamental Identities a)Sin²x + Cos²x = 1 b) Sum & Difference Formulas a)Cos (u – v) = Cos u.

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

0 1 - x Cos 0 1 - x Cos2

21 xCos 1 xCos

,3

x

3

50 x

Because we squared both sides, check your answers!


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 · 2 cot 4x 2 tan x + sec x x2 cos 2x 1 + sin x cos x tan 4 3x cos 2x f'(x) = (x + cos x)2 — + COS — (it) — 2 — + COS 2. 8. 11. 14. 20. 1 — sin — cos 3x y — —cosec

 · 2 cot 4x 2 tan x + sec x x2 cos 2x 1 + sin x cos x tan 4 3x cos 2x f'(x) = (x + cos x)2 — + COS — (it) — 2 — + COS 2. 8. 11. 14. 20. 1 — sin — cos 3x y — —cosec

Chapter 7 Answers - Welcome to CaltechAUTHORS - … · (x + ])sin x + cos x + C 3. x sin 5x/5 + cos 5x/25 + C 5. (x2 ... (x - 2)(x + 3l3l2 + C 33. x sin 3x/3 + cos 3x/9 + C 35. 3x

Chapter 7 Answers - Welcome to CaltechAUTHORS - … · (x + ])sin x + cos x + C 3. x sin 5x/5 + cos 5x/25 + C 5. (x2 ... (x - 2)(x + 3l3l2 + C 33. x sin 3x/3 + cos 3x/9 + C 35. 3x

Answers Part -II · 37. 1 1 4 cos(tan ) C 4 ... cos cos C 6 4 x x− + 6. 1 4 1 6 6 1 4 4 1 2 cos cos cosx x x− − 2 +C 7. 1 2 1 4 4 1 12 sin sinx x− 12 ...

Answers Part -II · 37. 1 1 4 cos(tan ) C 4 ... cos cos C 6 4 x x− + 6. 1 4 1 6 6 1 4 4 1 2 cos cos cosx x x− − 2 +C 7. 1 2 1 4 4 1 12 sin sinx x− 12 ...

Subtraction Formula for Cosine - Noorduyn's Math …...Compound Angles cos(x+ y) = cos x cos y — sin x sin y cos(x — y) = cos x cos y + sin x sin y 1. Express each of the following

Subtraction Formula for Cosine - Noorduyn's Math …...Compound Angles cos(x+ y) = cos x cos y — sin x sin y cos(x — y) = cos x cos y + sin x sin y 1. Express each of the following

sin .cos ∫ - MT Educare,mteducaresouth.com/images/maths/homework/Indefinite-Integration-13... · sin cos + sin .cos sin .cos x x dx ... = x2 + 3x +7 log |7x – 1|+ c Q-10) ...

sin .cos ∫ - MT Educare,mteducaresouth.com/images/maths/homework/Indefinite-Integration-13... · sin cos + sin .cos sin .cos x x dx ... = x2 + 3x +7 log |7x – 1|+ c Q-10) ...

s56f37ad12276e55c.jimcontent.com...tan-3 3x2 sec-2 3x2 2 cot 4x 4x 5x tan 5x2 X COS x + X 2X COS X-2 — 2X3 X 2 3X COS 3X — lot 2 sen 5t2 -+2 cos 5t2 2ax COS(aX2) — x tan 3x2

s56f37ad12276e55c.jimcontent.com...tan-3 3x2 sec-2 3x2 2 cot 4x 4x 5x tan 5x2 X COS x + X 2X COS X-2 — 2X3 X 2 3X COS 3X — lot 2 sen 5t2 -+2 cos 5t2 2ax COS(aX2) — x tan 3x2

PROJECTILE MOTION 3.1 Introduction. … · From equation (i) and (ii) 2 2 2 2 cos 1 cos sin u x g x y u 2 2 2 cos 1 tan u gx y x This equation shows that the trajectory of projectile

PROJECTILE MOTION 3.1 Introduction. … · From equation (i) and (ii) 2 2 2 2 cos 1 cos sin u x g x y u 2 2 2 cos 1 tan u gx y x This equation shows that the trajectory of projectile

hrsbstaff.ednet.ns.cahrsbstaff.ednet.ns.ca/macdonaz/IB Math SL Year 1 & 2/… ·  · 2013-12-09{quotient rule} cosm cosæ — sin sin x) cos a. cos sin x cos2 x sin2 + cos x = cos2

hrsbstaff.ednet.ns.cahrsbstaff.ednet.ns.ca/macdonaz/IB Math SL Year 1 & 2/… ·  · 2013-12-09{quotient rule} cosm cosæ — sin sin x) cos a. cos sin x cos2 x sin2 + cos x = cos2