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Created by T. Madas, , TRIGONOMETRY, THE DOUBLE ANGLE IDENTITIES, , Created by T. Madas

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Created by T. Madas, Question 2, Prove the validity of each of the following trigonometric identities., , a) cot 2 x + cosec 2 x ≡ cot x, b) cos 2 x + tan x sin 2 x ≡ 1, , c), , sin x, 1, ≡ cot x, 1 − cos x, 2, , d) sin 2θ ≡, , e), , 2 tan θ, 1 + tan 2 θ, , 1, 1, −, ≡ 2 sin θ sec 2θ, cos θ − sin θ cos θ + sin θ, , Created by T. Madas

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Created by T. Madas, Question 5, Prove the validity of each of the following trigonometric identities., , a), , 2 tan x, x, ≡ sec 2  , tan x + sin x, 2, , b) cot 2 x ≡, , cot 2 x − 1, 2 cot x, , 1, c) cosec θ − cot θ ≡ tan θ, 2, , d) 2 − 2 tan x −, , e), , 2 tan x, 2, ≡ (1 − tan x ), tan 2 x, , sin 2 x + sin x, ≡ tan x, cos 2 x + cos x + 1, , Created by T. Madas

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Created by T. Madas, Question 6, Prove the validity of each of the following trigonometric identities., , a), , 2 tan 2 x, ≡ cosec 2 x, tan 2 x − sin 2 x, , b), , sec 2 x − 1, ≡ tan 2 x, sec 2 x + 1, , c) tan A (1 + sec 2 A ) ≡ tan 2 A, , d), , cos 2 x − cos x + 1, ≡ cot x, sin 2 x − sin x, , e), , 1 + cos x, 1, ≡ cot 2 x, 1 − cos x, 2, , Created by T. Madas

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Created by T. Madas, Question 8, Prove the validity of each of the following trigonometric identities., , a) 4cosec2 2θ − sec2θ ≡ cosec2 θ, 1, b) 2 cos 4 θ + sin 2 2θ − 1 ≡ cos 2θ, 2, , c), , cos 2 x, ≡ cos x − sin x, 1 + sin 2 x, , d), , 2 − 2 cos x, x, ≡ sec, sin x, 2, , Created by T. Madas

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Created by T. Madas, Question 10, Solve each of the following trigonometric equations., , a) sin 2θ = tan θ ,, b) 2 sin 2 x = cos x ,, c) sin 2 y + sin y = 0 ,, d) 4sin ϕ cos ϕ = 1 ,, , 0 ≤ θ ≤ 180°, 0 ≤ x < 180°, 0 ≤ y < 360°, 0 ≤ϕ <π, , θ = 0°, 45°, 135°, 180° , x = 90°, x ≈14.5°, 165.5° , y = 0°, 120°, 180°, 240° ,, ϕ=, , Created by T. Madas, , π, , 5π, 12 12, ,

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Created by T. Madas, Question 13, Solve each of the following trigonometric equations., , a) 2 cos 2θ = 1 + cos θ ,, b) cos 2 x + 3sin x = 2 ,, c) cos 2 y + sin y = 0 ,, d) 2 (1 − cos 2ϕ ) = tan ϕ ,, , 0 ≤ θ < 360°, 0 ≤ x < 360°, 0 ≤ y < 360°, 0 ≤ ϕ < 180°, , θ = 0°, θ ≈ 138.6°, 221.4° , x = 30°, 90°, 150° , y = 90°, 210°, 330° ,, ϕ = 0°, 15°, 75°, , Created by T. Madas

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Created by T. Madas, Question 14, Solve each of the following trigonometric equations., , a) cos 2θ − 7sin θ − 4 = 0 ,, b) 3cos 2 x = sin x + 2 ,, c) 3cos 2 y = 7 cos y ,, d) cos 2ϕ = sin ϕ ,, , 0 ≤ θ < 360°, 0 ≤ x < 360°, , 0 ≤ y < 360°, 0 ≤ ϕ < 360°, , θ = 210°, 330° , x ≈ 19.5°, 160.5° x = 210°, 330° , y ≈ 109.5°, 250.5° ,, ϕ = 30°, 150°, 270°, , Created by T. Madas

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Created by T. Madas, Question 15, Solve each of the following trigonometric equations., , a) 3cos 2θ − 5sin θ = 4 ,, b) 3cos 2 x = 1 − sin x ,, c) cos 2 y − 7 cos y + 4 = 0 ,, d) cos 2ϕ + 6cos ϕ + 5 = 0 ,, , 0 ≤ θ < 360°, 0 ≤ x < 360°, 0 ≤ y < 360°, 0 ≤ ϕ < 360°, , θ = 210°, 330° θ ≈ 199.5°, 340.5° , x ≈ 41.8°, 138.2° x = 210°, 330° ,, y = 60°, 300° , ϕ = 180°, , Created by T. Madas

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Created by T. Madas, Question 16, Solve each of the following trigonometric equations., , a) cos 2θ = 7 cos θ + 3 ,, , 0 ≤ θ < 360°, , b) 2 cos 2 x = 4 cos x − 3 ,, c) 6 cos 2 y + 5cos y + 3 = 0 ,, d) 5cos 2ϕ + 22sin ϕ = 9 ,, , 0 ≤ x < 360°, 0 ≤ y < 360°, 0 ≤ ϕ < 360°, , θ = 120°, 240° , x = 60°, 300° , y ≈ 70.5°, 138.6°, 221.4°, 289.5° , ϕ ≈ 11.5°, 168.5°, , Created by T. Madas

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Created by T. Madas, Question 17, Solve each of the following trigonometric equations., , a) cos 2θ + 9sin θ + 4 = 0 ,, , 0 ≤ θ < 360°, , b) 3cos 2 x = 9 − 14 cos x ,, , 0 ≤ x < 360°, , c) 2 cos 2 y + 7 cos y = 0 ,, d) 2 cos 2ϕ = 1 − 2sin ϕ ,, , 0 ≤ y < 360°, 0 ≤ ϕ < 360°, , θ = 210°, 330° , x ≈ 48.2°, 311.8° , y ≈ 75.5°, 284.5° , ϕ = 54°, 126°, 198°, 342°, , Created by T. Madas

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Created by T. Madas, Question 18, Solve each of the following trigonometric equations., , a) cos 2θ = 1 + sin θ ,, , 0 ≤ θ < 360°, , b) cos 2 x + 3cos x = 1 ,, , 0 ≤ x < 2π, , c) 3cos 2 y = 1 − sin y ,, , 0 ≤ y < 360°, , 1 , d) 2 cos ϕ + 1 = sin  ϕ  ,, 2 , , 0 ≤ ϕ < 360°, , θ = 0°, 180°, 210°, 330° , x =, , π, , 5π, ,, 3 3, ,, , y ≈ 41.8°, 138.2°, , y = 210°, 330° ,, , ϕ = 97.2°, 262.8°, , Created by T. Madas

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Created by T. Madas, Question 20, Show clearly that each of the following trigonometric equations has no real roots, regardless of, the solution interval., , a) cos 2θ = 23 + 14 cos θ, b) cos 2 x + cos x + 2 = 0, proof, , Created by T. Madas