5.5 – Multiple-Angle and Product-to-Sum Identities

Παρουσίαση με θέμα: "5.5 – Multiple-Angle and Product-to-Sum Identities"— Μεταγράφημα παρουσίασης:

1 5.5 – Multiple-Angle and Product-to-Sum Identities
Part 2

2 Half-Angle Identities

3 Half-Angle Identities
sin Ɵ 2 = ± 1 − cos Ɵ 2

4 Half-Angle Identities
sin Ɵ 2 = ± 1 − cos Ɵ cos Ɵ 2 = ± cos Ɵ 2

5 Half-Angle Identities
sin Ɵ 2 = ± 1 − cos Ɵ 2 cos Ɵ 2 = ± 1+ cos Ɵ 2 tan Ɵ 2 = ± 1 − cos Ɵ 1+ cos Ɵ

6 Half-Angle Identities
sin Ɵ 2 = ± 1 − cos Ɵ 2 cos Ɵ 2 = ± 1+ cos Ɵ 2 tan Ɵ 2 = ± 1 − cos Ɵ 1+ cos Ɵ tan Ɵ 2 = 1 − cos Ɵ sin Ɵ

7 Half-Angle Identities
sin Ɵ 2 = ± 1 − cos Ɵ 2 cos Ɵ 2 = ± 1+ cos Ɵ 2 tan Ɵ 2 = ± 1 − cos Ɵ 1+ cos Ɵ tan Ɵ 2 = 1 − cos Ɵ sin Ɵ tan Ɵ 2 = sin Ɵ 1 + cos Ɵ

8 Ex. 1 Find the exact value of tan 7π 12

9 Ex. 1 Find the exact value of tan 7π 12 tan 7π 6 2

10 Ex. 1 Find the exact value of tan 7π 12 tan 7π 6 2 tan Ɵ 𝟐 = 𝐬𝐢𝐧 Ɵ 𝟏 + 𝐜𝐨𝐬 Ɵ

11 Ex. 1 Find the exact value of tan 7π 12 tan 7π 6 2 tan Ɵ 𝟐 = 𝐬𝐢𝐧 Ɵ 𝟏 + 𝐜𝐨𝐬 Ɵ sin 7π 6 1 + cos 7π 6

12 Ex. 1 Find the exact value of tan 7π 12 tan 7π 6 2 tan Ɵ 𝟐 = 𝐬𝐢𝐧 Ɵ 𝟏 + 𝐜𝐨𝐬 Ɵ sin 7π cos 7π 6 − − 3 2

13 Ex. 1 Find the exact value of tan 7π 12 tan 7π 6 2 tan Ɵ 𝟐 = 𝐬𝐢𝐧 Ɵ 𝟏 + 𝐜𝐨𝐬 Ɵ sin 7π cos 7π 6 − − 3 2 = − − 3 2

14 Ex. 1 Find the exact value of tan 7π 12 tan 7π 6 2 tan Ɵ 𝟐 = 𝐬𝐢𝐧 Ɵ 𝟏 + 𝐜𝐨𝐬 Ɵ sin 7π cos 7π 6 − − 3 2 = − − 3 2 = − 1 2 · 2 2 − 3

15 Ex. 1 Find the exact value of tan 7π 12 tan 7π 6 2 tan Ɵ 𝟐 = 𝐬𝐢𝐧 Ɵ 𝟏 + 𝐜𝐨𝐬 Ɵ sin 7π cos 7π 6 − − 3 2 = − − 3 2 = − 1 2 · 2 2 − 3 = 1 2 − 3

16 Ex. 1 Find the exact value of tan 7π 12 tan 7π 6 2 tan Ɵ 𝟐 = 𝐬𝐢𝐧 Ɵ 𝟏 + 𝐜𝐨𝐬 Ɵ sin 7π cos 7π 6 − − 3 2 = − − 3 2 = − 1 2 · 2 2 − 3 = 1 2 − 3 = -2 − 3

17 Ex. 2 Solve 2 sin 2 𝑥 2 + cos x = 1 + sin x on [0, 2π).

18 Ex. 2 Solve 2 sin 2 𝑥 2 + cos x = 1 + sin x on [0, 2π)
Ex. 2 Solve 2 sin 2 𝑥 2 + cos x = 1 + sin x on [0, 2π). 2 ± 1 − cos 𝑥 cos x = 1 + sin x

19 Ex. 2 Solve 2 sin 2 𝑥 2 + cos x = 1 + sin x on [0, 2π)
Ex. 2 Solve 2 sin 2 𝑥 2 + cos x = 1 + sin x on [0, 2π). 2 ± 1 − cos 𝑥 cos x = 1 + sin x 2 · 1 − cos 𝑥 2 + cos x = 1 + sin x

20 Ex. 2 Solve 2 sin 2 𝑥 2 + cos x = 1 + sin x on [0, 2π)
Ex. 2 Solve 2 sin 2 𝑥 2 + cos x = 1 + sin x on [0, 2π). 2 ± 1 − cos 𝑥 cos x = 1 + sin x 2 · 1 − cos 𝑥 2 + cos x = 1 + sin x 1 − cos 𝑥 + cos x = 1 + sin x

21 Ex. 2 Solve 2 sin 2 𝑥 2 + cos x = 1 + sin x on [0, 2π)
Ex. 2 Solve 2 sin 2 𝑥 2 + cos x = 1 + sin x on [0, 2π). 2 ± 1 − cos 𝑥 cos x = 1 + sin x 2 · 1 − cos 𝑥 2 + cos x = 1 + sin x 1 − cos 𝑥 + cos x = 1 + sin x sin x = 0

22 Ex. 2 Solve 2 sin 2 𝑥 2 + cos x = 1 + sin x on [0, 2π)
Ex. 2 Solve 2 sin 2 𝑥 2 + cos x = 1 + sin x on [0, 2π). 2 ± 1 − cos 𝑥 cos x = 1 + sin x 2 · 1 − cos 𝑥 2 + cos x = 1 + sin x 1 − cos 𝑥 + cos x = 1 + sin x sin x = 0 x = 0 and π

23 Product-to-Sum Identities

24 Product-to-Sum Identities
sin α sin β = ½[cos(α – β) – cos(α + β)]

25 Product-to-Sum Identities
sin α sin β = ½[cos(α – β) – cos(α + β)] cos α cos β = ½[cos(α – β) + cos(α + β)]

26 Product-to-Sum Identities
sin α sin β = ½[cos(α – β) – cos(α + β)] cos α cos β = ½[cos(α – β) + cos(α + β)] sin α cos β = ½[sin(α + β) + sin(α – β)]

27 Product-to-Sum Identities
sin α sin β = ½[cos(α – β) – cos(α + β)] cos α cos β = ½[cos(α – β) + cos(α + β)] sin α cos β = ½[sin(α + β) + sin(α – β)] cos α sin β = ½[sin(α + β) – sin(α – β)]

28 Sum-to-Product Identities

29 Sum-to-Product Identities
sin α + sin β = 2 sin α+ β 2 cos α – β 2

30 Sum-to-Product Identities
sin α + sin β = 2 sin α+ β 2 cos α – β 2 sin α – sin β = 2 cos α+ β 2 sin α – β 2

31 Sum-to-Product Identities
sin α + sin β = 2 sin α+ β 2 cos α – β 2 sin α – sin β = 2 cos α+ β 2 sin α – β 2 cos α + cos β = 2 cos α+ β 2 cos α – β 2

32 Sum-to-Product Identities
sin α + sin β = 2 sin α+ β 2 cos α – β 2 sin α – sin β = 2 cos α+ β 2 sin α – β 2 cos α + cos β = 2 cos α+ β 2 cos α – β 2 cos α – cos β = -2 sin α+ β 2 sin α – β 2

33 Ex. 3 Find the exact value of cos 7π 12 – cos π 12

34 Ex. 3 Find the exact value of cos 7π 12 – cos π 12 cos α – cos β = -2 sin 𝜶+ 𝜷 𝟐 sin 𝜶 – 𝜷 𝟐

35 Ex. 3 Find the exact value of cos 7π 12 – cos π 12 cos α – cos β = -2 sin 𝜶+ 𝜷 𝟐 sin 𝜶 – 𝜷 𝟐 = -2 sin 7π 12 + π 12 2 sin 7π 12 – π 12 2

36 Ex. 3 Find the exact value of cos 7π 12 – cos π 12 cos α – cos β = -2 sin 𝜶+ 𝜷 𝟐 sin 𝜶 – 𝜷 𝟐 = -2 sin 7π 12 + π 12 2 sin 7π 12 – π 12 2 = -2 sin 2π 3 2 sin π 2 2

37 Ex. 3 Find the exact value of cos 7π 12 – cos π 12 cos α – cos β = -2 sin 𝜶+ 𝜷 𝟐 sin 𝜶 – 𝜷 𝟐 = -2 sin 7π 12 + π 12 2 sin 7π 12 – π 12 2 = -2 sin 2π 3 2 sin π 2 2 = -2 sin π 3 sin π 4

38 Ex. 3 Find the exact value of cos 7π 12 – cos π 12 cos α – cos β = -2 sin 𝜶+ 𝜷 𝟐 sin 𝜶 – 𝜷 𝟐 = -2 sin 7π 12 + π 12 2 sin 7π 12 – π 12 2 = -2 sin 2π 3 2 sin π 2 2 = -2 sin π 3 sin π 4 = -2( 3 2 )( 2 2 )

39 Ex. 3 Find the exact value of cos 7π 12 – cos π 12 cos α – cos β = -2 sin 𝜶+ 𝜷 𝟐 sin 𝜶 – 𝜷 𝟐 = -2 sin 7π 12 + π 12 2 sin 7π 12 – π 12 2 = -2 sin 2π 3 2 sin π 2 2 = -2 sin π 3 sin π 4 = -2( 3 2 )( 2 2 ) = - 6 2

40 Ex. 4 Solve sin x + sin 5x = 0

41 Ex. 4 Solve sin x + sin 5x = 0 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐

42 Ex. 4 Solve sin x + sin 5x = 0 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐 2 sin 𝑥 + 5𝑥 2 cos 𝑥 – 5𝑥 2 = 0

43 Ex. 4 Solve sin x + sin 5x = 0 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐 2 sin 𝑥 + 5𝑥 2 cos 𝑥 – 5𝑥 2 = 0 2 sin 3x cos -2x = 0

44 Ex. 4 Solve sin x + sin 5x = 0 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐 2 sin 𝑥 + 5𝑥 2 cos 𝑥 – 5𝑥 2 = 0 2 sin 3x cos -2x = 0 2 sin 3x cos 2x = 0

45 Ex. 4 Solve sin x + sin 5x = 0 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐 2 sin 𝑥 + 5𝑥 2 cos 𝑥 – 5𝑥 2 = 0 2 sin 3x cos -2x = 0 2 sin 3x cos 2x = 0 sin 3x cos 2x = 0

46 Ex. 4 Solve sin x + sin 5x = 0 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐 2 sin 𝑥 + 5𝑥 2 cos 𝑥 – 5𝑥 2 = 0 2 sin 3x cos -2x = 0 2 sin 3x cos 2x = 0 sin 3x cos 2x = 0 sin 3x = 0 cos 2x = 0

47 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐
Ex. 4 Solve sin x + sin 5x = 0 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐 2 sin 𝑥 + 5𝑥 2 cos 𝑥 – 5𝑥 2 = 0 2 sin 3x cos -2x = 0 2 sin 3x cos 2x = 0 sin 3x cos 2x = 0 sin 3x = 0 cos 2x = 0 3x = 0 + 2nπ and π + 2nπ

48 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐
Ex. 4 Solve sin x + sin 5x = 0 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐 2 sin 𝑥 + 5𝑥 2 cos 𝑥 – 5𝑥 2 = 0 2 sin 3x cos -2x = 0 2 sin 3x cos 2x = 0 sin 3x cos 2x = 0 sin 3x = 0 cos 2x = 0 3x = 0 + 2nπ and π + 2nπ x = π nπ and 3π nπ

49 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐
Ex. 4 Solve sin x + sin 5x = 0 sin α + sin β = 2 sin 𝜶+ 𝜷 𝟐 cos 𝜶 – 𝜷 𝟐 2 sin 𝑥 + 5𝑥 2 cos 𝑥 – 5𝑥 2 = 0 2 sin 3x cos -2x = 0 2 sin 3x cos 2x = 0 sin 3x cos 2x = 0 sin 3x = 0 cos 2x = 0 3x = 0 + 2nπ and π + 2nπ x = π nπ and 3π nπ x = 2𝑛π 3 ; π 𝑛π 3 ; π 4 + nπ ; 3π 4 + nπ