Trigonometry

Higher Mathematics Unit 2 – Mathematics 25 Further Trigonometric EquationsWe will now consider trigonometric equations where the double-angleformulae are used to find solutions. These equations will involve: sin2x and either sin x or cos x cos2x and cos x cos2x and sin xSolving equations involving sin2x and either sinx or cosxEXAMPLE1. Solve sin2x° = − sin x° for 0 ≤ x < 360 .2sin x° cos x° = − sin x°2sin x° cos x° + sin x° = 0sin x° = 0sin x° ( 2cos x° + 1)= 02cos x° + 1 = 0 Replace sin2x using the double angleformula Take all terms to one side, making theequation equal to zero Factorise the expression and solvex = 0 or 180 or 360 cos x° = − 12x = 180 − 60 or 180 + 60So x = 0 or 120 or 180 or 240 .Solving equations involving cos2x and cosxEXAMPLE2. Solve cos2x= cos x for 0 ≤ x ≤ 2π.= 120 or 240RememberThe double-angleformulae are given inthe exam̌ S Ǎ T Cx = cos= 60−( )1 12cos2x= cos x22cos x − 1 = cos22cos x cos x 1 0− − =( 2cos x + 1)( cos x − 1)= 02cos x + 1 = 0cos x = − 12x = π − π3 or π + π3= 2π43 or π3So x = 0 or 2π 43 or π3 or 2π.x Replace cos2x by22cos x − 1 Take all terms to one side, making aquadratic equation in cos x Solve the quadratic equation (usingfactorisation or the quadratic formula)̌ S Ǎ T Cx = cos= π3−( )1 12cos x − 1 = 0cos x = 1x = 0 or 2πhsn.uk.netPage 99HSN22300