ADDITONAL MATHEMATICS

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1 ADDITONAL MATHEMATICS 00 0 CLASSIFIED TRIGONOMETRY Compiled & Edited B Dr. Eltaeb Abdul Rhman First Edition 0

2 5 Show that cosθ + + cosθ = cosec θ. [3] 0606//M/J/ 5 (i) x On the diagram above, sketch the curve = + 3sinx for 0º x 80º. [3] (ii) x On the diagram above, sketch the curve = + 3sinx for 0º x 80º. [] (iii) Write down the number of solutions of the equation + 3sinx = for 0º x 80º. [] 0606//M/J/

3 5 3 (a) O π π x The figure shows the graph of = k + m sin px for 0 x π, where k, m and p are positive constants. Complete the following statements. k =... m =... p =... [3] (b) The function g is such that g(x) = + 5cos3x. Write down (i) the amplitude of g, [] (ii) the period of g in terms of π. [] 0606//M/J/

4 3 Solve (i) cot x 5cosec x = 0 for 0 < x < 80, [5] (ii) 5cos 4sin = 0 for 0 < < 80, [4] (iii) cos z + π 6 = for 0 < z < π radians. [3] 0606//M/J/

5 0 (i) Solve the equation 3 sin x + 4 cos x = 0 for 0 < x < 360. [3] (ii) Solve the equation 6 cos + 6 sec = 3 for 0 < < 360. [5] (iii) Solve the equation sin(z 3) = 0.7 for 0 < z < π radians. [3] 0606//M/J/

6 5 4 (a) Prove that sin x = tan x sec x. [3] + sin x (b) An acute angle x is such that sin x = p. Given that sin x = sin x cos x, find an expression, in terms of p, for cosec x. [3] 0606//M/J/

7 8 6 (a) (q, ) O (0, p) x ( ϖ, 5) The diagram shows part of the graph = p + 3tan3x passing through the points ( π, 5), (0, p) and (q, ). Find the value of p and of q. [4] (b) It is given that f(x) = a cos(bx)+c, where a, b and c are integers. The maximum value of f is, the minimum value of f is 3 and the period of f is 7. Find the value of a, of b and of c. [4] 0606//O/N/

8 9 (i) Solve + cot x = 8 sin x for 0 x 360. [5] (ii) Solve 4 sin( 0.3) + 5 cos( 0.3) = 0 for 0 π radians. [5] (iii) Show that = sin θ cos θ. [3] tan θ + cot θ 0606//O/N/

9 3 0 (i) Solve sec x = 5 tan x + 5, for 0 < x < 360. [5] (ii) Solve sin + π =, 3 for 0 < < 4π radians. [5] 0606//O/N/

10 9 The function f is defined, for 0 x 360, b f(x) = sin 3x. 0 (i) State the amplitude and period of f. [] (ii) State the maximum value of f and the corresponding values of x. [3] (iii) Sketch the graph of f. [] 0606/3/O/N/

11 8 7 (a) Given that tan x = p, find an expression, in terms of p, for cosec x. [3] (b) Prove that ( + secθ )( cosθ ) = sinθ tanθ. [4] 6 (a) (i) On the same diagram, sketch the curves = cos x and = + cos x for 0 x π. [3] (ii) Hence state the number of solutions of the equation cos x cos x + = 0 where 0 x π. [] (b) The function f is given b f(x) = 5sin 3x. Find (i) the amplitude of f, [] (ii) the period of f. [] 0606//O/N/

12 4 (i) On the grid below, draw on the same axes, for 0 x 80, the graphs of = sin x and = + cos x. [3] (ii) State the number of roots of the equation sin x = + cos x for 0 x 80. [] (iii) Without extending our graphs state the number of roots of the equation sin x = + cos x for 0 x 360. [] 8 (a) Solve, for 0 x, the equation + 5cos 3x = 0, giving our answer in radians correct to decimal places. [3] (b) Find all the angles between 0º and 360º such that sec + 5tan = 3cos. [5] 0606//O/N/

13 (i) Solve 4 cot x =, for 0 x 360. [3] 0 The function f is defined, for 0 x 80, b f(x) = 3cos 4x. (i) Solve the equation f(x) = 0. [3] (ii) State the amplitude of f. [] (iii) State the period of f. [] (iv) State the maximum and minimum values of f. [] (v) Sketch the graph of = f(x). [3] UCLES /0/M/J/06 (ii) Solve 3( tan cos ) = 5 cos, for [5] 5 The function f is defined, for 0 x 80, b f(x) = A + 5 cos Bx, where A and B are constants. (i) Given that the maximum value of f is 3, state the value of A. [] (ii) State the amplitude of f. [] (iii) Given that the period of f is 0, state the value of B. [] (iv) Sketch the graph of f. [3] 0 (a) Given that a = sec x + cosec x and b = sec x cosec x, show that a + b sec x cosec x. [4] (b) Find, correct to decimal places, the values of between 0 and 6 radians which satisf the equation cot = 3sin. [5] 0606/3/O/N/

14 3 (iii) Solve 3 sec z = 4, for 0 z π radians. [3] 4 Prove the identit (! sec θ)(cosec θ 0 cot θ) ] tan θ. [4] 4 The functions f and g are defined b π f : x sin x, 0 x, g : x x 3, x. Solve the equation g f(x) = g (.75). [5] 4 The function f is defined, for 0 x 360, b f(x) = a sin (bx) + c, where a, b and c are positive integers. Given that the amplitude of f is and the period of f is 0, (i) state the value of a and of b. [] Given further that the minimum value of f is 0, (ii) state the value of c, [] (iii) sketch the graph of f. [3] 3 Given that θ is acute and that sinθ =, express, without using a calculator, sinθ cosθ sinθ in the 3 form a + b, where a and b are integers. [5] 4 (a) Given that sin x = p and cos x = p, where x is acute, find the exact value of p and the exact value of cosec x. [3] (b) Prove that (cot x + tan x) (cot x tan x) = sin x cos x. [3] 0606/3/O/N/

15 3 5 3 π π π π x The diagram shows part of the graph of = a tan (bx) + c. Find the value of (i) c, (ii) b, (iii) a. [3] (a) Solve, for 0 < x < 3 radians, the equation 4 sin x 3 = 0, giving our answers correct to decimal places. [3] (b) Solve, for 0 < < 360, the equation 4 cosec = 6 sin + cot. [6] Solve the equation (i) 3 sin x + 5 cos x = 0 for 0 < x < 360, [3] (ii) 3 tan sec = 0 for 0 < < 360, [5] (iii) sin(z 0.6) = 0.8 for 0 < z < 3 radians. [4] 6 (a) Given that sin x = p, find an expression, in terms of p, for sec x. [] (b) Prove that sec A cosec A cot A tan A. [4] 0 Solve (i) 4 sin x = cos x for 0 < x < 360, [3] (ii) 3 + sin = 3 cos for 0 < < 360, [5] (iii) sec z 3 = 4 for 0 < z < 5 radians. [3] 0606/0/M/J/07

16 5 Solve the equation (i) 3 sin x 4 cos x = 0, for 0 x 360, [3] (ii) sin + = 4 cos, for 0 360, [4] (iii) sec ( z + π 3 ) =, for 0 z π radians. [4] 5 (i) x On the diagram above, sketch the curve = + 3sinx for 0º x 80º. [3] (ii) x On the diagram above, sketch the curve = + 3sinx for 0º x 80º. [] (iii) Write down the number of solutions of the equation + 3sinx = for 0º x 80º. [] 0606//M/J/

17 5 3 (a) O π π x The figure shows the graph of = k + m sin px for 0 x π, where k, m and p are positive constants. Complete the following statements. k =... m =... p =... [3] (b) The function g is such that g(x) = + 5cos3x. Write down (i) the amplitude of g, [] (ii) the period of g in terms of π. [] 7 (a) Sets A and B are such that A = {x : sin x = 0.5 for 0 x 360 }, B = {x : cos (x 30 ) = 0.5 for 0 x 360 }. Find the elements of (i) A, [] (ii) A B. [] (b) Set C is such that C = {x : sec 3x = for 0 x 80 }. Find n(c). [3] 0606//M/J/

18 3 Solve (i) cot x 5cosec x = 0 for 0 < x < 80, [5] 8 The function f is defined, for 0 x π, b f(x) = sin x. State (i) the amplitude of f, [] (ii) the period of f, [] (iii) the maximum and minimum values of f. [] Sketch the graph of = f(x). [3] (ii) 5cos 4sin = 0 for 0 < < 80, [4] (iii) cos z + π 6 = for 0 < z < π radians. [3] x 3 The diagram shows part of the graph of = asin(bx) + c. State the value of (i) a, (ii) b, (iii) c. [3] 0606//M/J/

19 0 (i) Solve the equation 3 sin x + 4 cos x = 0 for 0 < x < 360. [3] (ii) Solve the equation 6 cos + 6 sec = 3 for 0 < < 360. [5] (a) Solve, for 0 x 360, the equation cot x = + tan x. [5] (b) Given that is measured in radians, find the two smallest positive values of such that 6sin( + ) + 5 = 0. [5] 9 (a) Find all the angles between 0 and 360 which satisf the equation 3cosx = 8tan x. [5] (b) Given that 4 6, find the value of for which 3 3 cos + = 0. [3] (iii) Solve the equation sin(z 3) = 0.7 for 0 < z < π radians. [3] 4 The function f is given b f : x + 5 sin 3x for 0 x 80. (i) State the amplitude and period of f. [] (ii) Sketch the graph of = f(x). [3] 0606//M/J/

20 4 (a) Prove that 5 sin x = tan x sec x. [3] + sin x 3 Show that cos sec = sin. [4] 9 (a) Solve, for 0 ` x ` 360, the equation 4 tan x! 8 sec x #. [4] (b) Given that ` 4, find the largest value of such that 5 tan(! ) # 6. [4] Prove the identit cos x cot x + sin x cosec x. [4] 6 The function f is defined, for 0 ` x ` π, b f (x) # 5! 3 cos 4x. Find (i) the amplitude and the period of f, [] (ii) the coordinates of the maximum and minimum points of the curve # f (x). [4] (b) An acute angle x is such that sin x = p. Given that sin x = sin x cos x, find an expression, in terms of p, for cosec x. [3] 9 (a) Solve, for 0 x 360, the equation sin x # 3 cos x! 4 sin x. [4] (b) Solve, for 0 ` ` 4, the equation cot # 0.5, giving our answers in radians correct to decimal places. [4] 6 Given that x = 3sinθ cosθ and = 3cosθ + sinθ, (i) find the value of the acute angle θ for which x =, [3] (ii) show that x + is constant for all values of θ. [3] 0606//M/J/

21 6 (i) State the amplitude of + sin ( x 3 ). [] (ii) State, in radians, the period of + sin ( x 3 ). [] A B =.5 = + sin x ( 3) O The diagram shows the curve = + sin ( x 3 ) meeting the line =.5 at points A and B. Find x (iii) the x-coordinate of A and of B, [3] (iv) the area of the shaded region. [6] 3 Show that cos x sin x + cos x + sin x = sec x. [4] 5 A curve has the equation = x sin x + π 3. The curve passes through the point P ( π, a ). (i) Find, in terms of π, the value of a. [] (ii) Using our value of a, find the equation of the normal to the curve at P. [5]

22 3 Show that cosθ + + cosθ = cosec θ. [3] Solve the equation (i) 5 sin x 3 cos x = 0, for 0 x 360, [3] (ii) cos sin = 0, for 0 360, [5] 3 (iii) 3 sec z = 0, for 0 z 6 radians. (a) [3] º 60º 0º 80º 40º 300º 360º x The diagram shows the curve = A cos Bx + C for 0 x 360. Find the value of (i) A, (ii) B, (iii) C. [3] (b) Given that f(x) = 6 sin x + 7, state (i) the period of f, [] (ii) the amplitude of f. [] 0606/3/O/N/0

23 ADDITONAL MATHEMATICS 00 0 CLASSIFIED TRIGONOMETRY Compiled & Edited B Dr. Eltaeb Abdul Rhman First Edition 0

ADDITONAL MATHEMATICS

ADDITONAL MATHEMATICS 2002 2011 CLASSIFIED REMAINDER THEOREM Compiled & Edited By Dr. Eltayeb Abdul Rhman www.drtayeb.tk First Edition 2011 7 5 (i) Show that 2x 1 is a factor of 2x 3 5x 2 + 10x 4. [2]