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PSfrag replacements O x y Higher Mathematics Trig. Past Papers Unit 2 Outcome 3 Written Questions 5 1. Solve the equation 3 cos 2x ◦ + cos x ◦ = −1 in the interval 0 ≤ x ≤ 360. [SQA] Part •1 •2 •3 •4 •5 Marks 5 ss: pd: ss: pd: pd: Level A/B Calc. CR Content T10 know to use cos 2x = 2 cos2 x − 1 process know to/and factorise quadratic process process Answer 60, 131·8, 228·2, 300 •1 •2 •3 •4 •5 U2 OC3 2000 P2 Q5 3(2 cos2 x ◦ − 1) 6 cos2 x ◦ + cos x ◦ − 2 = 0 (2 cos x ◦ − 1)(3 cos x ◦ + 2) cos x ◦ = 12 , x = 60, 30 cos x ◦ = − 23 , x = 132, 228 2. Solve the equation cos 2x ◦ + 5 cos x ◦ − 2 = 0, 0 ≤ x < 360. [SQA] 5 frag replacements O x y 3. Find the exact solutions of the equation 4 sin2 x = 1, 0 ≤ x < 2π . [SQA] frag replacements O x y replacements O x y hsn.uk.net Page 1 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes 4 PSfrag replacements O x y Higher Mathematics 4. Solve the equation 2 cos2 x = 12 , for 0 ≤ x ≤ π . [SQA] 3 frag replacements O x y 5 5. Solve the equation cos 2x ◦ + cos x ◦ = 0, 0 ≤ x < 360. [SQA] frag replacements O x y 4 6. Solve 2 sin 3x ◦ − 1 = 0 for 0 ≤ x ≤ 180. [SQA] frag replacements O x y replacements O x y hsn.uk.net Page 2 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements O x y Higher Mathematics 7. [SQA] (a) Show that 2 cos 2x ◦ − cos2 x ◦ = 1 − 3 sin2 x ◦ . 2 (b) Hence solve the equation 2 cos 2x ◦ − cos2 x ◦ = 2 sin x ◦ in the interval 0 ≤ x < 360. 4 frag replacements O x y 5 8. Solve the equation sin 2x ◦ + sin x ◦ = 0, 0 ≤ x < 360. [SQA] frag replacements O x y 9. Find, correct to one decimal place, the value of x between 180 and 270 which satisfies the equation 3 cos(2x ◦ − 40◦ ) − 1 = 0. [SQA] frag replacements O x y replacements O x y hsn.uk.net Page 3 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes 5 PSfrag replacements O x y Higher Mathematics 10. [SQA] (a) Write the equation cos 2θ + 8 cos θ + 9 = 0 in terms of cos θ and show that, for cos θ , it has equal roots. 3 (b) Show that there are no real roots for θ . 1 frag replacements O x y 11. If f (a) = 6 sin2 a − cos a, express f (a) in the form p cos 2 a + q cos a + r . [SQA] Hence solve, correct to three decimal places, the equation 6 sin2 a − cos a = 5 for 0 ≤ a ≤ π. 4 frag replacements O x y 12. Find the values of t, where 0 < t < 2π , for which 4 cos 2t − value. [SQA] π 4 has its maximum frag replacements replacements O x y O x y hsn.uk.net Page 4 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes 4 PSfrag replacements O x y Higher Mathematics 13. Solve the equation 2 sin 2x − [SQA] π 6 4 = 1, 0 ≤ x < 2π . frag replacements O x y 14. (a) Solve the equation sinPSfrag 2x ◦ − cos x ◦ = 0 in the interval 0 ≤ x ≤ 180. replacements y y = sin 2x ◦ (b) The diagram shows parts of two trigonometric graphs, y = sin 2x ◦ and y = cos x ◦ . 180 x 90 O Use your solutions in (a) to write down the coordinates of the point P. y = cos x ◦ P [SQA] Part (a) (b) •1 •2 •3 •4 Marks 4 1 ss: pd: pd: pd: •5 ic: Level C C Calc. NC NC Content T10 T3 Answer 30, 90, 150 √ (150, − 23 ) use double angle formula factorise process process or interpret graph •1 •2 •3 •4 2 sin x ◦ cos x ◦ cos x ◦ (2 sin x ◦ − 1) cos x ◦ = 0, sin x ◦ = 90, 30, 150 U2 OC3 2001 P1 Q5 1 2 •3 sin x ◦ = 12 and x = 30, 150 •4 cos x ◦ = 0 and x = 90 √ •5 150, − 23 replacements O x y hsn.uk.net Page 5 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes 4 1 PSfrag replacements O x y Higher Mathematics 15. [SQA] frag replacements O x y frag replacements O x y 16. Functions f and g are defined on suitable domains by f (x) = sin(x ◦ ) and g(x) = 2x . [SQA] (a) Find expressions for: (i) f (g(x)); 2 (ii) g( f (x)). 5 (b) Solve 2 f (g(x)) = g( f (x)) for 0 ≤ x ≤ 360. Part (a) (b) Marks 2 5 •1 ic: •2 ic: Level C C Calc. CN CN Content A4 T10 Answer (i) sin(2x ◦ ), (ii) 2 sin(x ◦ ) 0◦ , 60◦ , 180◦ , 300◦ , 360◦ ss: equate for intersection ss: substitute for sin 2x pd: extract a common factor pd: solve a ‘common factor’ equation •7 pd: solve a ‘linear’ equation •3 •4 •5 •6 •7 2 sin(2x ◦ ) = 2 sin(x ◦ ) appearance of 2 sin(x ◦ ) cos(x ◦ ) 2 sin(x ◦ ) (2 cos(x ◦ ) − 1) sin(x ◦ ) = 0 and 0, 180, 360 cos(x ◦ ) = 12 and 60, 300 or •6 sin(x ◦ ) = 0 and cos(x ◦ ) = •7 0, 60, 180, 300, 360 replacements hsn.uk.net 2002 P1 Q3 •1 sin(2x ◦ ) •2 2 sin(x ◦ ) interpret f (g(x)) interpret g( f (x)) •3 •4 •5 •6 O x y U2 OC3 Page 6 1 2 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements O x y Higher Mathematics 17. [SQA] frag replacements O x y frag replacements O x y replacements O x y hsn.uk.net Page 7 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements PSfrag replacements Higher Mathematics 18. The diagram shows the graph of a cosine function from 0 to π . [SQA] (a) State the equation of the graph. √ (b) The line with equation y = − 3 intersects this graph at point A and B. Marks 1 3 •1 ic: Level C C Calc. NC NC y 2 1 O π 2 A B −2 Find the coordinates of B. Part (a) (b) O x y Content T4 T7 interpret graph •2 ss: equate equal parts •3 pd: solve linear trig equation in radians 4 • ic: interpret result Answer y = 2 cos √2x 7π B( 12 , − 3) x √ y=− 3 π U2 OC3 2002 P1 Q8 •1 2 cos 2x √ •1 2 cos 2x = − 3 7π •2 2x = 5π 6 , 6 7π •3 x = 12 19. [SQA] frag replacements O x y frag replacements O x y replacements O x y hsn.uk.net Page 8 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes 3 PSfrag replacements O x y Higher Mathematics 20. [SQA] frag replacements O x y frag replacements O x y replacements O x y hsn.uk.net Page 9 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements O x y Higher Mathematics 21. [SQA] frag replacements O x y frag replacements O x y replacements O x y hsn.uk.net Page 10 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements O x y Higher Mathematics 22. [SQA] frag replacements O x y frag replacements O x y 23. [SQA] frag replacements O x y frag replacements O x y replacements O x y hsn.uk.net Page 11 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements O x y Higher Mathematics 24. [SQA] frag replacements O x y frag replacements O x y replacements O x y hsn.uk.net Page 12 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements O x y Higher Mathematics 25. [SQA] frag replacements O x y frag replacements O x y 26. If cos θ = 45 , 0 ≤ θ < [SQA] π 2, find the exact value of (a) sin 2θ 2 (b) sin 4θ . 3 frag replacements O x y 27. Given that tan α = [SQA] √ 11 3 , 0<α< π 2, find the exact value of sin 2α. frag replacements replacements O x y O x y hsn.uk.net Page 13 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes 3 PSfrag replacements O x y Higher Mathematics 28. Given that cos D = [SQA] √2 5 and 0 < D < π 2, find the exact values of sin D and cos 2D . 3 frag replacements O x y 29. Given that sin A = 34 , where 0 < A < [SQA] π 2, find the exact value of sin 2A. 3 frag replacements O x y 30. For acute angles P and Q, sin P = [SQA] 12 13 and sin Q = 35 . Show that the exact value of sin(P + Q) is 63 65 . 3 frag replacements O x y replacements O x y hsn.uk.net Page 14 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements O x y Higher Mathematics 31. Find the exact value of sin θ ◦ + sin(θ ◦ + 120◦ ) + cos(θ ◦ + 150◦ ). [SQA] 3 frag replacements O x y 32. If x ◦ is an acute angle such that tan x ◦ = √ 4 3+3 sin(x ◦ + 30◦ ) is . 10 [SQA] 4 3, show that the exact value of 3 frag replacements O x y replacements O x y hsn.uk.net Page 15 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements O x y Higher Mathematics 33. [SQA] frag replacements O x y frag replacements O x y 34. A and B are acute angles such that tan A = [SQA] 3 4 and tan B = 5 12 . Find the exact value of (a) sin 2A 2 (b) cos 2A 1 (c) sin(2A + B). 2 frag replacements O x y replacements O x y hsn.uk.net Page 16 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements O x y Higher Mathematics 35. Functions f (x) = sin x , g(x) = cos x and h(x) = x + set of real numbers. [SQA] π 4 are defined on a suitable (a) Find expressions for: (i) f (h(x)); 2 (ii) g(h(x)). (b) (i) Show that f (h(x)) = √1 2 sin x + √1 2 cos x . (ii) Find a similar expression for g(h(x)) and hence solve the equation f (h(x)) − g(h(x)) = 1 for 0 ≤ x ≤ 2π . Part (a) Marks 2 Level C Calc. NC (b) 5 C NC •1 ic: •2 ic: •3 •4 •5 •6 •7 ss: ic: ic: pd: pd: Content A4 T8, T7 interpret composite functions interpret composite functions Answer (i) sin(x + π4 ), (ii) cos(x + π4 ) (i) proof, (ii) x = π4 , 3π 4 •1 sin(x + π4 ) •2 cos(x + π4 ) •3 sin x cos π4 complete •4 g(h(x)) = expand sin(x + π4 ) interpret substitute start solving process process + cos x sin π4 5 U2 OC3 2001 P1 Q7 and √1 cos x − √1 sin x 2 2 •5 ( √12 sin x + √12 cos x) − ( √12 cos x − √12 •6 √22 sin x •7 x = π4 , 3π accept only radians 4 replacements O x y hsn.uk.net Page 17 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes sin x) PSfrag replacements O x y Higher Mathematics y 36. On the coordinate diagram shown, A is the PSfrag replacements point (6, 8) and B is the point (12, −5). Angle AOC = p and angle COB = q. [SQA] A(6, 8) 4 Find the exact value of sin(p + q). O p C q x B(12, −5) Part •1 •2 •3 •4 Marks 4 ss: pd: ic: pd: Level C Calc. NC Content T9 Answer U2 OC3 63 65 2000 P1 Q1 know to use trig expansion •1 sin p cos q + cos p sin q process missing sides •2 10 and 13 PSfrag replacements 8 12 6 5 interpret data •3 10 · 13 + 10 · 13 126 4 O process • 130 x y 37. In triangle ABC, show that the exact 2 value of sin(a + b) is √ . 5 [SQA] C a A Part •1 •2 •3 •4 Marks 4 pd: ss: pd: pc: Level C Calc. NC Content T9 process the missing sides expand substitute process and complete proof 4 1 1 3 Answer proof b B U2 OC3 2002 P1 Q5 √ √ •1 AC = 2 and BC = 10 stated or implied by •3 2 • sin(a + b) = sin a cos b + cos a sin b •3 √12 · √310 + √12 · √110 •4 √4 20 = ... = √2 5 replacements O x y hsn.uk.net Page 18 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements O x y Higher Mathematics 38. [SQA] frag replacements O x y frag replacements O x y replacements O x y hsn.uk.net Page 19 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes PSfrag replacements O x y Higher Mathematics 39. [SQA] frag replacements O x y frag replacements O x y [END OF WRITTEN QUESTIONS] replacements O x y hsn.uk.net Page 20 c SQA Questions marked ‘[SQA]’ c Higher Still Notes All others