C3 Exam Workshop 2. Workbook. 1. (a) Express 7 cos x 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2

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1 C3 Exam Workshop 2 Workbook 1. (a) Express 7 cos x 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2 π. Give the value of α to 3 decimal places. (b) Hence write down the minimum value of 7 cos x 24 sin x. (c) Solve, for 0 x < 2π, the equation (3) (1) 7 cos x 24 sin x = 10, giving your answers to 2 decimal places. (5) BlueStar Mathematics Workshops (2011) 1

2 2. (a) Express 1.5 sin 2x + 2 cos 2x in the form R sin (2x + α), where R > 0 and 1 0 < α < 2 π, giving your values of R and α to 3 decimal places where appropriate. (4) (b) Express 3 sin x cos x + 4 cos 2 x in the form a cos 2x + b sin 2x + c, where a, b and c are constants to be found. (2) (c) Hence, using your answer to part (a), deduce the maximum value of 3 sin x cos x + 4 cos 2 x. (2) BlueStar Mathematics Workshops (2011) 2

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4 3. (a) Prove that 1 cos 2θ tan θ, θ sin 2θ (b) Solve, giving exact answers in terms of π, n π, n Z. 2 (3) 2(1 cos 2θ ) = tan θ, 0 < θ < π. (6) BlueStar Mathematics Workshops (2011) 4

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6 4. (a) Express sin x + 3 cos x in the form R sin (x + α), where R > 0 and 0 < α < 90. (4) (b) Show that the equation sec x + 3 cosec x = 4 can be written in the form sin x + 3 cos x = 2 sin 2x. (c) Deduce from parts (a) and (b) that sec x + 3 cosec x = 4 can be written in the form sin 2x sin (x + 60 ) = 0. X + Y X Y (d) Hence, using the identity sin X sin Y = 2 cos sin, or otherwise, 2 2 find the values of x in the interval 0 x 180, for which sec x + 3 cosec x = 4. (5) (3) (1) BlueStar Mathematics Workshops (2011) 6

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8 5. (i) (a) Express (12 cos θ 5 sin θ) in the form R cos (θ + α), where R > 0 and 0 < α < 90. (b) Hence solve the equation (4) 12 cos θ 5 sin θ = 4, for 0 < θ < 90, giving your answer to 1 decimal place. (3) (ii) Solve 8 cot θ 3 tan θ = 2, for 0 < θ < 90, giving your answer to 1 decimal place. (5) BlueStar Mathematics Workshops (2011) 8

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10 6. (i) Given that sin x = 5 3, use an appropriate double angle formula to find the exact value of sec 2x. (ii) Prove that cot 2x + cosec 2x cot x, nπ x, Z 2 n. (4) (4) BlueStar Mathematics Workshops (2011) 10

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12 7. (a) Prove that 2 1 tan θ cos 2θ tan θ (4) (b) Hence, or otherwise, prove tan 2 π = (5) BlueStar Mathematics Workshops (2011) 12

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14 8. (a) Prove that for all values of x 2 tan x sin 2x = 2sin 2 x tan x. (5) (b) Hence, or otherwise, find the values of x in the interval 0 x 360º, for which 2 tan x sin2x = sin 2 x giving your answers to an appropriate degree of accuracy. (6) BlueStar Mathematics Workshops (2011) 14

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16 9. (a) Using the half-angle formulae, or otherwise, prove that for all values of x 1+ cos x 1 cos x x cot2 2. (5) (b) Hence, or otherwise, find the values of x in the interval 0 x 2π for which 1+ cos x 1 cos x = 6cosec x 2 10 (7) BlueStar Mathematics Workshops (2011) 16

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18 10. (a) Prove that there are no real values of θ for which cos2θ + cosθ + 2 = 0. (b) Find the values of x in the interval 0 x 360º, for which 3sin x 2cos 2 x = 0 (c) Hence, find the values of y in the interval 0 y 180º, for which 3sec2y 2 cot 2y = 0 (4) (5) (4) BlueStar Mathematics Workshops (2011) 18

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20 11. (a) Prove that for all values of x ( ). cos 2 x sin 2 2x cos 2 x 4 cos 2 x 3 (5) (b) Hence, or otherwise, find the values of x in the interval 0 x 2π for which cos 2 x sin 2 2x = 0 giving your answer in terms of π. (6) BlueStar Mathematics Workshops (2011) 20

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22 12. (a) Show that for all values of x, where x is measured in degrees, cos( x + 60 ) 3sin( x 60 ) 2 cos x 3sin x. (5) (b) Hence, find the values of x in the interval -180º x 180º, for which cos( x + 60º ) 3sin( x 60º ) = 0 (4) BlueStar Mathematics Workshops (2011) 22

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24 13. (a) Using the identity cos (A + B) cos A cos B sin A sin B, prove that (b) Show that cos 2A 1 2 sin 2 A. (2) 2 sin 2θ 3 cos 2θ 3 sin θ + 3 sin θ (4 cos θ + 6 sin θ 3). (c) Express 4 cos θ + 6 sin θ in the form R sin (θ + α ), where R > 0 and 0 < α < π 2 1. (d) Hence, for 0 θ < π, solve (4) (4) 2 sin 2θ = 3(cos 2θ + sin θ 1), giving your answers in radians to 3 significant figures, where appropriate. (5) BlueStar Mathematics Workshops (2011) 24

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26 14. (a) Use the double angle formulae and the identity cos(a + B) cosa cosb sina sinb to obtain an expression for cos 3x in terms of powers of cos x only. (4) (b) (i) Prove that cos x 1+ sin x + 1+ sin x cos x 2 sec x, x (2n + 1) 2 π. (4) (ii) Hence find, for 0 < x < 2π, all the solutions of cos x 1+ sin x + 1+ sin x cos x = 4. (3) BlueStar Mathematics Workshops (2011) 26

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28 15. a) Given that sin( θ + α ) = 2.5sinθ, show that tanθ = sinα 2.5 cosα. (3) b) Hence, solve the equation sin( θ + 45 ) = 2.5sinθ, given 0 θ 360. (4) END BlueStar Mathematics Workshops (2011) 28