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Algebra II Trigonometry Honors Unit 10: Day 9 Notes: Double Angle Identities Classwork Find the exact value of each. 3 where 270 ≤ θ < 360 5 Find sin 2θ 2) cos θ = 3 where 630 ≤ θ < 720 5 Find tan 2θ 4 5π where 2π ≤ θ < 3 2 Find cos 2θ 4) sin θ = 1) cos θ = 12 π where 0 ≤ θ < 13 2 Find tan 2θ 3) tan θ = Verify each identity. 5) 2tan 2 xsin xcos x = sin 2x 6) 2 cot x 2sin xcos x = tan 2x cos 2x 7) cos 2 x(1 − cos 2x) = sin 2 x(1 + cos 2x) 8) sec 2 x − 2sin 2 x = cos 2x + tan 2 x 9) tan 2 xcos 2 x = cos 2 x − cos 2x tan 2 x sin 2 x 10) = 1 − tan 2 x cos 2x 11) tan 2 x 1 = 1 − cos 2x 2cos 2 x 12) sin 2x + tan xcos x = sin x ⋅ (2cos x + 1) 13) 2cot x ⋅ (1 − cos 2x) = 2sin xcos x + sin 2x 14) 15) sin 2xsec 2 x 2 tan x = 2csc 2 xsin xcos x 1 − tan 2 x sec 2 x = sin 2x tan 2x 16) 2(sin xcos xcsc 2 x − 1) = sin 2x − 1 + cos 2x sin 2 x Worksheet by Kuta Software LLC ©v _2t0s1e7[ HKGuTtSam pSoobfKtQwLabrSeO vLsLlCz.] w ZAOl[l] irbipgbhYtgsQ MrdefsLeJrrvteadK.V V GMLaudCeV AwbiqtwhH bIGnvfZimndibt\e^ SPtrzeJcAajlOc`uRlquzsO. Answers to Classwork 1) − 24 25 2) 2 5) 2tan xsin xcos x 2sin xcos x cot 2 x sin 2x cot 2 x 24 7 Use cot x = 7 25 2sin xcos x 6) cos 2x 3) − 1 tan x 4) − Use sin 2x = 2sin xcos x Use sin 2x = 2sin xcos x sin 2x cos 2x Use tan 2x = ■ tan 2x ■ 2 7) cos x(1 − cos 2x) (1 − cos 2x)(1 + cos 2x) 2 sin 2 x(1 + cos 2x) 2 2 8) sec x − 2sin x Use cos 2 x = 1 + cos 2x 2 Use sin 2 x = 1 − cos 2x 2 120 119 sin 2x cos 2x ■ Use tan 2 x + 1 = sec 2 x tan 2 x + 1 − 2sin 2 x Use cos 2x = 1 − 2sin 2 x cos 2x + tan 2 x ■ 2 2 9) tan xcos x ( ) sin x cos x Decompose into sine and cosine 2 ⋅ cos 2 x Simplify sin 2 x Use cos 2x = cos 2 x − sin 2 x cos 2 x − cos 2x ■ 2 10) tan x Decompose into sine and cosine 1 − tan 2 x ( ) ( ) sin x cos x 2 sin x cos x 1− 2 sin 2 x 2 2 cos x − sin x sin 2 x cos 2x Simplify Use cos 2x = cos 2 x − sin 2 x ■ Worksheet by Kuta Software LLC ©h R2K0y1S7e eKOuftAaz ^SloffYtUwea]rKeJ xLgLICv.F S JAMlAlE FrliZgnhwtfsL crve\sfehrJvlejdt.l f XMnaxdHeR OwWiytjhf lIEnJf\iunTittBey EPOreeEczaelmcxumlYupsC. 11) tan 2 x 1 − cos 2x Use tan 2 x = 1 − cos 2x (1 − cos 2x)(1 + cos 2x) Cancel common factors 1 1 + cos 2x Use cos 2x = 2cos 2 x − 1 1 ■ 2cos 2 x 12) sin 2x + tan xcos x Use sin 2x = 2sin xcos x cos x ⋅ (2sin x + tan x) ( cos x ⋅ 2sin x + sin x cos x Decompose into sine and cosine ) sin x ⋅ (2cos x + 1) 13) 2cot x ⋅ (1 − cos 2x) 14) 1 − cos 2x 1 + cos 2x Simplify ■ Use cos 2x = 1 − 2sin 2 x cos x sin x 4cot xsin 2 x Use cot x = 4cos xsin 2 x sin x Cancel common factors 4sin xcos x Use sin 2x = 2sin xcos x 2sin xcos x + sin 2x sin 2xsec 2 x ■ tan 2 x ( ) ( ) 1 sin 2x ⋅ cos x sin x cos x Decompose into sine and cosine 2 2 sin 2x sin 2 x 2sin xcos x 2 sin x 2csc 2 xsin xcos x Simplify Use sin 2x = 2sin xcos x Use csc x = 1 sin x ■ Worksheet by Kuta Software LLC ©w g2D0Q1[7U xKCu_tGaF aScojfVtowOa^rle\ jL`LRCT.t u aAclKln RrRiYgdhNtXsG VrqensgeZrkvtegds.K f sM]a\ddet bwIiqtQhC GIsnOfkivnvittYeU SPKrlemcMaQlScluolruGsc. 15) 1 − tan 2 x sin 2x ( ) sin x 1− cos x sin 2x Decompose into sine and cosine 2 Simplify cos 2 x − sin 2 x cos 2 xsin 2x cos 2x Use cos 2x = cos 2 x − sin 2 x Use sec x = 2 cos xsin 2x sec 2 xcos 2x sin 2x Use tan 2x = sec 2 x ■ tan 2x 2 16) 2(sin xcos xcsc x − 1) ( ( ) ) 1 2 sin xcos x ⋅ sin x 2sin x ⋅ (cos x − sin x) sin 2 x sin 2x − 2sin 2 x 2 sin x sin 2x − 1 + cos 2x sin 2x cos 2x Decompose into sine and cosine 2 2(−sin x + cos x) sin x sin 2 x 1 cos x −1 Simplify Create a common factor Use sin 2x = 2sin xcos x Use cos 2x = 1 − 2sin 2 x ■ Worksheet by Kuta Software LLC ©v o2S0O1Q7Q qKkuXtjac zSao_fstnw`aprAeU QLXLTCw.] a bAYlWlf PrHipgNhvtzsc brZeXs\eGrkvNeidW.j N hMBaydpeL AwAihtEhr hIhnZfHiRngi\tMeF [PprFeRcuaultcPu`lIuVsW.