No Calc. 1 p (seen anywhere) 1 p. 1 p. No Calc. (b) Find an expression for cos 140. (c) Find an expression for tan (a) (i) sin 140 = p A1 N1

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Dennis Norman
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1 IBSL /4 IB REVIEW: Trig KEY (0points). Let p = sin 40 and q = cos 0. Give your answers to the following in terms of p and/or q. (a) Write down an expression for (i) sin 40; (ii) cos 70. (b) Find an expression for cos 40. (c) Find an expression for tan 40.. (a) (i) sin 40 = p N () () (ii) cos 70 = q N (b) METHOD evidence of using sin + cos = diagram, cos 40 = cos 40 = METHOD p (seen anywhere) p p N evidence of using cos = cos cos 40 = cos 70 cos 40 = ( q) (= q ) N (c) METHOD tan 40 = sin 40 cos 40 p p N METHOD tan 40 = p q N. Consider g (x) = sin x. (a) Write down the period of g. (b) Sketch the curve of g, for 0 x. (c) Write down the number of solutions to the equation g (x) =, for 0 x. () (). (a) period = N

2 (b) y 4 0 π 4 π π π x Note: Award for amplitude of, for their period, for a sine curve passing through (0, 0) and (0, ). N (c) evidence of appropriate approach line y = on graph, discussion of number of solutions in the domain 4 (solutions) N [6] π. (a) Given that cos A = and 0 A, find cos A. (b) Given that sin B = and π B, find cos B.. (a) evidence of choosing the formula cos A = cos A Note: If they choose another correct formula, do not award the M unless there is evidence of finding sin A =. correct substitution 8 cos A =, cos A 7 cosa N (b) METHOD evidence of using sin B + cos B = cos B, (seen anywhere), cos B = cos B = N

3 METHOD diagram M for finding third side equals cos B = N 4. Let f : x sin x. (a) (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. (b) Find f (x), giving your answer in the form a sin p x cos q x where a, p, q. π (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. 4. (a) (i) range of f is [, ], ( f (x) ) A N [6] () () (7) (ii) sin x = sin x = justification for one solution on [0, ] R e. g. x, unit circle, sketch of sin x solution (seen anywhere) N (b) f (x) = sin x cos x AN b a (c) using V y dx V sin x cos x dx 0 0 sin x cos x dx sin x 0 sin sin 0 V A evidence of using sin and sin 0 = 0 ( 0) V = N

4 π. The diagram below shows a circle centre O, with radius r. The length of arc ABC is cm and A ÔC =. Calc Ok (a) Find the value of r. (b) Find the perimeter of sector OABC. (c) Find the area of sector OABC. () () (). (a) evidence of appropriate approach M = r r =. (cm) N (b) adding two radii plus perimeter = 7+ (cm) (= 6.4) N (c) evidence of appropriate approach M. area = 0. (cm ) (= 6.6) N [6] 6. The following diagram shows a semicircle centre O, diameter [AB], with radius. Let P be a point on the circumference, with P ÔB = radians. (a) Find the area of the triangle OPB, in terms of. (b) Explain why the area of triangle OPA is the same as the area triangle OPB. Let S be the total area of the two sments shaded in the diagram below. ()

5 (c) Show that S = ( sin ). (d) Find the value of when S is a local minimum, justifying that it is a minimum. (e) Find a value of for which S has its greatest value. (8) () 6. (a) evidence of using area of a triangle A sinθ A = sin N (b) METHOD P ÔA = area OPA = sin θ (= sin ( )) since sin ( ) = sin R then both triangles have the same area AGN0 METHOD triangle OPA has the same height and the same base as triangle OPB then both triangles have the same area (c) area semi-circle = area APB = sin + sin (= 4 sin ) S = area of semicircle area APB (= 4 sin ) S = ( sin ) R AGN0 M AGN0 (d) METHOD attempt to differentiate ds 4cos θ dθ setting derivative equal to 0 correct equation 4 cos = 0, cos = 0, 4 cos = 0 = N EITHER evidence of using second derivative S() = 4 sin

6 S 4 it is a minimum because S 0 OR evidence of using first derivative R N0 for <, S () < 0 (may use diagram) for >, S () > 0 (may use diagram) it is a minimum since the derivative goes from native to positive RN0 METHOD 4 sin is minimum when 4 sin is a maximum R 4 sin is a maximum when sin = (A) = A N (e) S is greatest when 4 sin is smallest (or equivalent) (R) = 0 (or ) N 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. Calc Ok [8] (a) 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. (b) 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. (6) (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. () 7. (a) (i) 7 N (ii) N (iii) 0 N

7 (b) (i) evidence of appropriate approach M 8 A A = 8 AG N0 (ii) C = 0 AN (iii) METHOD period = evidence of using B period = (accept 60) = B B accept 0.4or0 N 6 METHOD evidence of substituting 0 = 8 cos B + 0 simplifying cos B = 0 B B accept 0.4or0 N 6 (c) correct answers t =., t = 0., between 0: and 0: (accept 0:0) N []

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

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