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Trigonometry

1.       (a)     Sketch the graph of f (x) = sin 3x + sin 6x, 0 ≤ x ≤ 2p.

(b)     Write down the exact period of the function f.

Working:

 

 

Answer:

(b)       ..................................................................

(Total 3 marks)

 


 

2.       The area of the triangle shown below is 2.21 cm2.  The length of the shortest side is x cm and the other two sides are 3x cm and (x + 3) cm.

(a)     Using the formula for the area of the triangle, write down an expression for sin θ in terms of x.

(2)

(b)     Using the cosine rule, write down and simplify an expression for cos θ in terms of x.

(2)

(c)     (i)      Using your answers to parts (a) and (b), show that,

(1)

(ii)     Hence find

(a)     the possible values of x;

(2)

(b)     the corresponding values of θ, in radians, using your answer to part (b) above.

(3)

(Total 10 marks)

 


 

3.       In a triangle ABC,  = 30°, AB = 6 cm and AC =  cm. Find the possible lengths of [BC].

Working:

 

 

Answer:

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(Total 3 marks)

 


 

4.       (a)     Let y = , where 0 < a < b.

(i)      Show that = .

(4)

(ii)     Find the maximum and minimum values of y.

(4)

(iii)    Show that the graph of y = , 0 < a < b cannot have a vertical asymptote.

(2)

(b)     For the graph of for 0 £ x £ 2p,

(i)      write down the y-intercept;

(ii)     find the x-intercepts m and n, (where m < n) correct to four significant figures;

(iii)    sketch the graph.

(5)

(c)     The area enclosed by the graph of  and the x-axis from x = 0 to x = n is denoted by A.  Write down, but do not evaluate, an expression for the area A.

(2)

(Total 17 marks)

 


 

5.       (a)     Sketch the graph of the function

 

for –2p £ x £ 2p

(5)

(b)     Prove that the function C(x) is periodic and state its period.

(3)

(c)     For what values of x, –2p £ x £ 2p, is C(x) a maximum?

(2)

(d)     Let x = x0 be the smallest positive value of x for which C(x) = 0.  Find an approximate value of x0 which is correct to two significant figures.

(2)

(e)     (i)      Prove that C(x) = C(–x) for all x.

(2)

(ii)     Let x = x1 be that value of x, p < x < 2p, for which C(x) = 0. Find the value of x1 in terms of x0.

(2)

(Total 16 marks)

 

 

6.       Solve 2 sin x = tan x, where  < x <

Working:

 

 

Answer:

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(Total 3 marks)

 


 

7.       Let θ be the angle between the unit vectors a and b, where 0 < θ < π. Express |ab| in terms of .

Working:

 

 

Answer:

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(Total 3 marks)

 

 

8.       The angle q satisfies the equation tanq + cotq  = 3, where q is in degrees. Find all the possible values of q lying in the interval [0°, 90°].

Working:

 

 

Answer:

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(Total 6 marks)

 


 

9.       The function f is defined on the domain [0, p] by f (q ) = 4 cos q + 3 sin q.

(a)     Express f (q ) in the form R cos (qa) where 0 < a < .

(b)     Hence, or otherwise, write down the value of q for which f (q ) takes its maximum value.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 6 marks)

 

 

10.     Triangle ABC has AB = 8 cm, BC = 6 cm and  = 20°. Find the smallest possible area of DABC.

Working:

 

 

Answer:

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(Total 6 marks)

 


 

11.     Find all the values of θ in the interval [0, p] which satisfy the equation

cos 2q  = sin2 q .

Working:

 

 

Answer:

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(Total 6 marks)

 

 

12.     In the triangle ABC,  = 30°, BC = 3 and AB = 5. Find the two possible values of .

Working:

 

 

Answer:

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(Total 6 marks)

 


 

13.     The function f with domain  is defined by f (x) = cos x + sin x.

          This function may also be expressed in the form R cos (xa) where R > 0 and 0 < α < .

(a)     Find the exact value of R and of α.

(3)

(b)     (i)      Find the range of the function f.

(ii)     State, giving a reason, whether or not the inverse function of f exists.

(5)

(c)     Find the exact value of x satisfying the equation f (x) =

(3)

(d)     Using the result

          = ln½sec x + tan x½+ C, where C is a constant,

          show that

         

(5)

(Total 16 marks)

 

 

14.     (a)     Express  cosq  – sinq  in the form r cos (q + a), where r > 0 and 0 < α < , giving r and α as exact values.

(3)

(b)     Hence, or otherwise, for 0 £ q  £ 2p, find the range of values of  cosq  – sinq.

(2)

(c)     Solve – sinq  = –l, for 0 £ q £ 2p, giving your answers as exact values.

(5)

(Total 10 marks)

 


 

15.     Prove that  = tan q, for 0 < q < , and q  ¹ .

(Total 5 marks)

 

 

16.     (a)     Show that cos (A + B) + cos(AB) = 2 cos A cos B

(2)

(b)     Let Tn (x) = cos (n arccos x) where x is a real number, x Î [–1, 1] and n is a positive integer.

(i)      Find T1(x).

(ii)     Show that T2 (x) = 2x2 – 1.

(5)

(c)     (i)      Use the result in part (a) to show that Tn+1 (x) + Tn–1 (x) = 2xTn (x).

(ii)     Hence or otherwise, prove by induction that Tn (x) is a polynomial of degree n.

(12)

(Total 19 marks)

 

 

17.     The diagram below shows a circle centre O and radius OA = 5 cm. The angle  = 135°.


 

          Find the area of the shaded region.

Working:

 

 

Answer:

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(Total 6 marks)

 

 

18.     The angle θ satisfies the equation 2 tan2 θ – 5 sec θ – 10 = 0, where θ is in the second quadrant. Find the exact value of sec θ.

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(Total 6 marks)

 


 

19.     A farmer owns a triangular field ABC. The side [AC] is 104 m, the side [AB] is 65 m and the angle between these two sides is 60°.

(a)     Calculate the length of the third side of the field.

(3)

(b)     Find the area of the field in the form p, where p is an integer.

(3)

          Let D be a point on [BC] such that [AD] bisects the 60° angle. The farmer divides the field into two parts by constructing a straight fence [AD] of length x metres.

(c)     (i)      Show that the area of the smaller part is given by  and find an expression for the area of the larger part.

(ii)     Hence, find the value of x in the form q, where q is an integer.

(8)

(d)     Prove that .

(6)

(Total 20 marks)

 

 

20.     Consider the complex geometric series eiθ e2iθ +e3iθ + …

(a)     Find an expression for z, the common ratio of this series.

(2)

(b)     Show that ½z½< 1.

(2)

(c)     Write down an expression for the sum to infinity of this series.

(2)

(d)     (i)      Express your answer to part (c) in terms of sin θ and cos θ.

(ii)     Hence show that

cos θ +cos 2θ + cos 3θ +…=

(10)

(Total 16 marks)

 

 

21.     Given that a sin 4x + b sin 2x = 0, for 0 < x < , find an expression for cos2 x in terms of a and b.

(Total 6 marks)

 

 

22.     The triangle ABC has an obtuse angle at B, BC = 10.2,  = x and  = 2x.

(a)     Find AC, in terms of cos x.

(b)     Given that the area of triangle ABC is 52.02 cos x, find angle .

(Total 6 marks)

 


 

23.     The diagram shows a trapezium OABC in which OA is parallel to CB. O is the centre of a circle radius r cm. A, B and C are on its circumference. Angle  = θ.

          Let T denote the area of the trapezium OABC.

(a)     Show that T =  (sin θ + sin 2θ).

(4)

          For a fixed value of r, the value of T varies as the value of θ varies.

(b)     Show that T takes its maximum value when θ satisfies the equation
4 cos2 θ + cos θ 2 = 0, and verify that this value of T is a maximum.

(5)

(c)     Given that the perimeter of the trapezium is 75 cm, find the maximum value of T.

(6)

(Total 15 marks)

 


 

24.     In triangle ABC,  = 31°, AC = 3 cm and BC = 5 cm. Calculate the possible lengths of the side [AB].

(Total 6 marks)

 


 

25.     The following diagram shows the points A and B on the circumference of a circle, centre O, and radius 4 cm, where  = q. Points A and B are moving on the circumference so that q is increasing at a constant rate.

          Given that the rate of change of the length of the minor arc AB is numerically equal to the rate of change of the area of the shaded segment, find the acute value of q.

(Total 6 marks)

 


 

26.     In the obtuse-angled triangle ABC, AC = 10.9 cm, BC = 8.71 cm and  = 50°.

Find the area of triangle ABC.

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(Total 6 marks)

 


 

27.     Let z = cos q + i sin q, for – < q < .

(a)     (i)      Find z3 using the binomial theorem.

(ii)     Use de Moivre’s theorem to show that

          cos 3q = 4 cos3q – 3 cosq and sin 3q = 3 sinq – 4 sin3q.

(10)

(b)     Hence prove that  = tanq.

(6)

(c)     Given that sinq = , find the exact value of tan 3q.

(5)

(Total 21 marks)

 


 

28.     Solve tan2 2q =1, in the interval

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(Total 6 marks)

 


 

29.     The graph below represents y = a sin (x + b) + c, where a, b, and c are constants.

Find values for a, b and c.

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(Total 6 marks)

 


 

30.     In the triangle ABC,  = 30°, a = 5 and c = 7. Find the difference in area between the two possible triangles for ABC.

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(Total 6 marks)

 


 

31.     Let    be the angles of a triangle. Show that tan  + tan  + tan  =  tan  tan

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(Total 6 marks)

 


 

32.     The following diagram shows a circle centre O, radius r. The angle  at the centre of the circle is q radians. The chord AB divides the circle into a minor segment (the shaded region) and a major segment.

(a)     Show that the area of the minor segment is r2 (q – sin q).

(4)

(b)     Find the area of the major segment.

(3)

(c)     Given that the ratio of the areas of the two segments is 2:3, show that sinq =q.

(4)

(d)     Hence find the value of q.

(2)

(Total 13 marks)

 


 

33.     Triangle ABC has = 42°, BC =1.74 cm, and area 1.19 cm2.

(a)     Find AC.

(b)     Find AB.

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(Total 6 marks)

 


 

34.     The diagram shows a circle centre O and radius 1, with  = q, q  0. The area of ΔAOB is three times the shaded area.

Find the value of q.

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(Total 6 marks)

 


 

35.     The following diagram shows DABC, where BC =105 m,  = 40°,  = 60°.

Find AB.

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(Total 6 marks)

 


 

36.     (a)     Using the formula for cos (A + B) prove that cos2q =

(3)

(b)     Hence, find

(4)

Let f (x) = cos x and g (x) = sec x for xÎ .

Let R be the region enclosed by the two functions.

(c)     Find the exact values of the x-coordinates of the points of intersection.

(4)

(d)     Sketch the functions f and g and clearly shade the region R.

(3)

The region R is rotated through 2p about the x-axis to generate a solid.

(e)     (i)      Write down an integral which represents the volume of this solid.

(ii)     Hence find the exact value of the volume.

(10)

(Total 24 marks)

 


 

37.     The diagram below shows a pair of intersecting circles with centres at P and Q with radii of 5 cm and 6 cm respectively. RS is the common chord of both circles and PQ is 7 cm.

Find the area of the shaded region.

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(Total 6 marks)

 


 

38.     The function f is defined by f (x) = cosec x + tan 2x.

(a)     Sketch the graph of f for

Hence state

(i)      the x-intercepts;

(ii)     the equations of the asymptotes;

(iii)    the coordinates of the maximum and minimum points.

(8)

(b)     Show that the roots of f (x) = 0 satisfy the equation

2 cos3x − 2 cos2x − 2 cos x + 1 = 0.

(5)

(c)     Show that the x-coordinates of the maximum and minimum points on the

curve satisfy the equation 4 cos5 x − 4 cos3 x + 2 cos2 x + cos x − 2 = 0.

(8)

(d)     Show that f (px) + f (p + x) = 0.

(4)

(Total 25 marks)

 


 

39.     In triangle ABC, AB = 9 cm, AC =12 cm, and  is twice the size of .

Find the cosine of .

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(Total 5 marks)

 


 

40.     In the diagram below, AD is perpendicular to BC.

CD = 4, BD = 2 and AD = 3.  = a and  = b.

Find the exact value of cos (ab).

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(Total 6 marks)

 


 

41.     The diagram below shows the boundary of the cross-section of a water channel.

          The equation that represents this boundary is y = 16 sec – 32 where x and y are both measured in cm.

          The top of the channel is level with the ground and has a width of 24 cm. The maximum depth of the channel is 16 cm.

          Find the width of the water surface in the channel when the water depth is 10 cm. Give your answer in the form a arccos b where a, bÎ.

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(Total 6 marks)

 

 

42.     A system of equations is given by

                           cos x + cos y = 1.2

                           sin x + sin y = 1.4.

(a)     For each equation express y in terms of x.

(2)

(b)     Hence solve the system for 0 < x < p, 0 < y < p.

(4)

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(Total 6 marks)

 


 

43.     Find, in its simplest form, the argument of (sinq + i (1− cosq ))2 where q is an acute angle.

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(Total 7 marks)

 


 

44.     In triangle ABC, BC = a, AC = b, AB = c and [BD] is perpendicular to [AC].

(a)     Show that CD = bc cos A.

(1)

(b)     Hence, by using Pythagoras’ Theorem in the triangle BCD, prove the cosine rule for the triangle ABC.

(4)

If  = 60°, use the cosine rule to show that c =

(7)

(Total 12 marks)

 

 

45.    

          The above three dimensional diagram shows the points P and Q which are respectively west and south-west of the base R of a vertical flagpole RS on horizontal ground. The angles of elevation of the top S of the flagpole from P and Q are respectively 35° and 40°, and PQ = 20 m.

Determine the height of the flagpole.

(Total 8 marks)

 


 

46.     The depth, h (t) metres, of water at the entrance to a harbour at t hours after midnight on a particular day is given by

          h (t) = 8 + 4 sin

(a)     Find the maximum depth and the minimum depth of the water.

(3)

(b)     Find the values of t for which h (t) ³ 8.

(3)

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(Total 6 marks)

 


 

47.     Consider triangle ABC with  = 37.8°, AB = 8.75 and BC = 6.

Find AC.

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(Total 7 marks)

 

 

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