Find the range of values of θ between 0 and 2 π for which cos 2 θ > cos θ .

Given: cos 2 θ > cos θ ⇒ 2 cos 2 θ - 1 > cos θ ⇒ 2 cos 2 θ - cos θ - 1 > 0 ⇒ ( 2 cos θ + 1 ) ( cos θ - 1 ) > 0 ⇒ cos θ < - 1 2 or cos θ > 1 The second inequality has no solutions. cos θ < - 1 2 Critical values are: cos θ = - 1 2 θ = 2 π 3 or θ = 2 π - 2 π 3 = 4 π 3 Thus, range is 2 π 3 < θ < 4 π 3

Use the expansions of cos ( 2 x + x ) and cos ( 2 x - x ) , to prove that: cos 3 x + cos x ≡ 2 cos 2 x cos x Solve cos 3 x + cos 2 x + cos x > 0 for 0 ° < x < 360 ° .

Take cos 3 x = cos ( 2 x + x ) = cos 2 x cos x - sin 2 x sin x ...(i) cos x = cos ( 2 x - x ) = cos 2 x cos x + sin 2 x sin x ..............(ii) (i) + (ii) cos 3 x + cos x = 2 cos 2 x cos x Hence, proved. Using cos 3 x + cos x = 2 cos 2 x cos x ⇒ cos 3 x + cos 2 x + cos x > 0 ⇒ ( cos 3 x + cos x ) + cos 2 x > 0 ⇒ 2 cos 2 x cos x + cos 2 x > 0 ⇒ cos 2 x ( 2 cos x + 1 ) > 0 ⇒ cos 2 x > 0 and cosx > - 1 2 ....(iii) or cos 2 x < 0 and cosx < - 1 2 ....(iv) Now, cos x > - 1 2 when x < 120 ° or x > 240 ° cos x < - 1 2 when 120 ° < x < 240 ° cos 2 x > 0 when 0 ° < 2 x < 90 ° or 270 ° < 2 x < 360 ° i.e., when 0 ° < x < 45 ° or 135 ° < x < 180 ° cos 2 x < 0 when 45 ° < x < 135 ° Here, (iii) requires x < 120 ° and 0 ° < x < 45 ° so 0 ° < x < 45 ° Here, (iv) requires 120 ° < x < 240 ° and 45 ° < x < 135 ° , so 120 ° < x < 135 °

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Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 3 - Trigonometry>EXERCISE 3C

Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Trigonometry, Exercise 4: EXERCISE 3C

Author:Sue Pemberton, Julianne Hughes & Julian Gilbey

Sue Pemberton Mathematics Solutions for Exercise - Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Trigonometry, Exercise 4: EXERCISE 3C

Attempt the free practice questions on Chapter 3: Trigonometry, Exercise 4: EXERCISE 3C with hints and solutions to strengthen your understanding. Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book solutions are prepared by Experienced Embibe Experts.