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

Half-Angle Identities

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

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

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

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

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

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

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

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

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

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 − 1 2 1 − 3 2

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 − 1 2 1 − 3 2 = − 1 2 2 − 3 2

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 − 1 2 1 − 3 2 = − 1 2 2 − 3 2 = − 1 2 · 2 2 − 3

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 − 1 2 1 − 3 2 = − 1 2 2 − 3 2 = − 1 2 · 2 2 − 3 = 1 2 − 3

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 − 1 2 1 − 3 2 = − 1 2 2 − 3 2 = − 1 2 · 2 2 − 3 = 1 2 − 3 = -2 − 3

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

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 2 2 + cos x = 1 + sin x 2 · 1 − cos 𝑥 2 + cos x = 1 + sin x

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 2 2 + cos x = 1 + sin x 2 · 1 − cos 𝑥 2 + cos x = 1 + sin x 1 − cos 𝑥 + cos x = 1 + sin x

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 2 2 + 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

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 2 2 + 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

Product-to-Sum Identities

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

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)]

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)]

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)]

Sum-to-Product Identities

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

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

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

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

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

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

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

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

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

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 )

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

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

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

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

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

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

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

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

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 2x = π 2 + 2nπ and 3π 2 + 2nπ

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 2x = π 2 + 2nπ and 3π 2 + 2nπ x = 2𝑛π 3 ; π 3 + 2𝑛π 3 ; π 4 + nπ ; 3π 4 + nπ