(b) Show that sin 2 =. 9 (c) Find the exact value of cos 2. (Total 6 marks)

Transcription

1 IB SL Trig Review. In the triangle PQR, PR = 5 cm, QR = 4 cm and PQ = 6 cm. Calculate the size of PQˆ R ; the area of triangle PQR.. The following diagram shows a triangle ABC, where AĈB is 90, AB =, AC = and BÂC is. 5 Show that sin =. 4 5 Show that sin =. 9 (c) Find the exact value of cos.. The following diagram shows a sector of a circle of radius r cm, and angle at the centre. The perimeter of the sector is 0 cm. 0 r Show that =. r The area of the sector is 5 cm. Find the value of r. 4. The following diagram shows the triangle AOP, where OP = cm, AP = 4 cm and AO = cm. A diagram not to scale O P Calculate AÔP, giving your answer in radians. The following diagram shows two circles which intersect at the points A and B. The smaller circle C has centre O and radius cm, the larger circle C has centre P and radius 4 cm, and OP = cm. The point D lies on the circumference of C and E on the circumference of C.Triangle AOP is the same as triangle AOP in the diagram above.

2 A C C D E O P diagram not to scale B Find AÔB, giving your answer in radians. (c) Given that APˆB is.6 radians, calculate the area of (i) sector PAEB; (ii) sector OADB. (d) The area of the quadrilateral AOBP is 5.8 cm. (i) Find the area of AOBE. (ii) Hence find the area of the shaded region AEBD. (Total 4 marks) 5. The following diagram shows a pentagon ABCDE, with AB = 9. cm, BC =. cm, BD = 7. cm, AÊD =0, ADˆ E = 5 and ABˆ D = 60. Find AD. Find DE. (c) The area of triangle BCD is 5.68 cm. Find DBˆ C. (d) (e) Find AC. Find the area of quadrilateral ABCD. (Total marks)

3 x 6. The diagram below shows the graph of f (x) = + tan for 60 x 60. On the same diagram, draw the asymptotes. Write down (i) the period of the function; (ii) the value of f (90 ). (c) Solve f (x) = 0 for 60 x Consider the equation 4x + kx + = 0. For what values of k does this equation have two equal roots? Let f be the function f ( ) = cos + 4 cos +, for Show that this function may be written as f ( ) = 4 cos + 4 cos +. (c) Consider the equation f ( ) = 0, for (i) How many distinct values of cos satisfy this equation? (ii) Find all values of which satisfy this equation. (d) Given that f ( ) = c is satisfied by only three values of, find the value of c. () (Total marks)

4 8. A Ferris wheel with centre O and a radius of 5 metres is represented in the diagram below. Initially seat A is at ground level. The next seat is B, where AÔB =. 6 Find the length of the arc AB. Find the area of the sector AOB. (c) The wheel turns clockwise through an angle of. Find the height of A above the ground. The height, h metres, of seat C above the ground after t minutes, can be modelled by the function h (t) = 5 5 cos t. 4 (d) (i) Find the height of seat C when t =. 4 (ii) Find the initial height of seat C. (iii) Find the time at which seat C first reaches its highest point. (8) (e) Find h (t). (f) For 0 t, (i) sketch the graph of h ; (ii) find the time at which the height is changing most rapidly. 9. Let p = sin 40 and q = cos 0. Give your answers to the following in terms of p and/or q. Write down an expression for (i) sin 40 ; (ii) cos 70. Find an expression for cos 40. (c) Find an expression for tan 40. (Total marks) () 4

5 0. Consider g (x) = sin x. Write down the period of g. On the diagram below, sketch the curve of g, for 0 x. y x () (c) Write down the number of solutions to the equation g (x) =, for 0 x.. Given that cos A = and 0 A, find cos A. Given that sin B = and B, find cos B.. Let f : x sin x. (i) Write down the range of the function f. (ii) Consider f (x) =, 0 x. Write down the number of solutions to this equation. Justify your answer. Find f (x), giving your answer in the form a sin p x cos q x where a, p, q. (c) Let g (x) = sin x(cos x) for 0 x. Find the volume generated when the curve of g is revolved through about the x-axis.. The following diagram shows a semicircle centre O, diameter [AB], with radius. Let P be a point on the circumference, with PÔB = radians. (7) (Total 4 marks) Find the area of the triangle OPB, in terms of. Explain why the area of triangle OPA is the same as the area triangle OPB. Let S be the total area of the two segments shaded in the diagram below. 5

6 (c) Show that S = ( sin ). (d) (e) Find the value of when S is a local minimum, justifying that it is a minimum. Find a value of for which S has its greatest value. 4. The diagram below shows triangle PQR. The length of [PQ] is 7 cm, the length of [PR] is 0 cm, and is 75. (8) (Total 8 marks) PQˆ R Find PQˆ R. Find the area of triangle PQR. 5. The diagram below shows a circle centre O, with radius r. The length of arc ABC is cm and AÔC =. 9 Find the value of r. Find the perimeter of sector OABC. (c) Find the area of sector OABC. 6

7 6. Let f (x) = 4 tan x 4 sin x, x. On the grid below, sketch the graph of y = f (x). Solve the equation f (x) =. 7. The following graph shows the depth of water, y metres, at a point P, during one day. The time t is given in hours, from midnight to noon. Use the graph to write down an estimate of the value of t when (i) the depth of water is minimum; (ii) the depth of water is maximum; (iii) the depth of the water is increasing most rapidly. The depth of water can be modelled by the function y = A cos (B (t )) + C. (i) Show that A = 8. (ii) Write down the value of C. (iii) Find the value of B. (c) A sailor knows that he cannot sail past P when the depth of the water is less than m. Calculate the values of t between which he cannot sail past P. (Total marks) (6) 7