PSf Written Questions Trig. Past Papers Unit utcome 3 1. Solve the equation 3 cos + cos = 1 in the interval 0 360. 5 Part Marks Level Calc. Content Answer U C3 5 A/B CR T10 60, 131 8, 8, 300 000 P Q5 1 ss: know to use cos = cos 1 pd: process 3 ss: know to/and factorise quadratic 4 pd: process 5 pd: process 1 3( cos 1) 6 cos + cos = 0 3 ( cos 1)(3 cos + ) 4 cos = 1, = 60, 30 5 cos = 3, = 13, 8. Solve the equation cos + 5 cos = 0, 0 < 360. 5 3. Find the eact solutions of the equation 4 sin = 1, 0 < π. 4 hsn.uk.net Page 1 Questions marked c SQA
PSf 4. Solve the equation cos = 1, for 0 π. 3 5. Solve the equation cos + cos = 0, 0 < 360. 5 6. Solve sin 3 1 = 0 for 0 180. 4 hsn.uk.net Page Questions marked c SQA
PSf 7. (a) Show that cos cos = 1 3 sin. (b) Hence solve the equation cos cos = sin in the interval 0 < 360. 4 8. Solve the equation sin + sin = 0, 0 < 360. 5 9. Find, correct to one decimal place, the value of between 180 and 70 which satisfies the equation 3 cos( 40 ) 1 = 0. 5 hsn.uk.net Page 3 Questions marked c SQA
PSf 10. (a) Write the equation cos θ + 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 11. If f (a) = 6 sin a cos a, epress f (a) in the form p cos a + q cos a + r. Hence solve, correct to three decimal places, the equation 6 sin a cos a = 5 for 0 a π. 4 1. Find the values of t, where 0 < t < π, for which 4 cos ( t π ) 4 has its maimum value. 4 hsn.uk.net Page 4 Questions marked c SQA
PSf 13. ( ) Solve the equation sin π 6 = 1, 0 < π. 4 14. (a) Solve the equation sinpsfrag cos = 0 in the interval 0 180. 4 = sin (b) The diagram shows parts of two trigonometric graphs, = sin and = cos. 180 Use our solutions in (a) to write 90 down the coordinates of the point P. 1 P = cos Part Marks Level Calc. Content Answer U C3 (a) 4 C NC T10 30, 90, 150 001 P1 Q5 (b) 1 C NC T3 (150, 3 ) 1 ss: use double angle formula pd: factorise 3 pd: process pd: process 4 1 sin cos cos ( sin 1) 3 cos = 0, sin = 1 4 90, 30, 150 5 ic: interpret graph or 3 sin = 1 and = 30, 150 4 cos = 0 and = 90 (150, 5 ) 3 hsn.uk.net Page 5 Questions marked c SQA
PSf 15. 16. Functions f and g are defined on suitable domains b f () = sin( ) and g() =. (a) Find epressions for: (i) f (g()); (ii) g( f ()). (b) Solve f (g()) = g( f ()) for 0 360. 5 Part Marks Level Calc. Content Answer U C3 (a) C CN A4 (i) sin( ), (ii) sin( ) 00 P1 Q3 (b) 5 C CN T10 0, 60, 180, 300, 360 1 ic: interpret f (g()) ic: interpret g( f ()) 3 ss: equate for intersection 4 ss: substitute for sin 5 pd: etract a common factor 6 pd: solve a common factor equation pd: solve a linear equation 7 hsn.uk.net Page 6 or 1 sin( ) sin( ) 3 sin( ) = sin( ) 4 appearance of sin( ) cos( ) 5 sin( ) ( cos( ) 1) 6 sin( ) = 0 and 0, 180, 360 7 cos( ) = 1 and 60, 300 6 sin( ) = 0 and cos( ) = 1 7 0, 60, 180, 300, 360 Questions marked c SQA
PSf 17. hsn.uk.net Page 7 Questions marked c SQA
PSf PSf 18. The diagram shows the graph of a cosine function from 0 to π. (a) State the equation of the graph. 1 (b) The line with equation = 3 π π intersects this graph at point A A B and B. = 3 Find the coordinates of B. 3 Part Marks Level Calc. Content Answer U C3 (a) 1 C NC T4 = cos 00 P1 Q8 (b) 3 C NC T7 B( 7π 1, 3) 1 ic: interpret graph 1 cos ss: equate equal parts 3 pd: solve linear trig equation in radians 4 ic: interpret result 1 cos = 3 = 5π 6, 7π 6 3 = 7π 1 19. hsn.uk.net Page 8 Questions marked c SQA
PSf 0. hsn.uk.net Page 9 Questions marked c SQA
PSf 1. hsn.uk.net Page 10 Questions marked c SQA
PSf. 3. hsn.uk.net Page 11 Questions marked c SQA
PSf 4. hsn.uk.net Page 1 Questions marked c SQA
PSf 5. 6. If cos θ = 4 5, 0 θ < π, find the eact value of (a) sin θ (b) sin 4θ. 3 7. Given that tan α = 11 3, 0 < α < π, find the eact value of sin α. 3 hsn.uk.net Page 13 Questions marked c SQA
PSf 8. Given that cos D = and 0 < D < π 5, find the eact values of sin D and cos D. 3 9. Given that sin A = 3 4, where 0 < A < π, find the eact value of sin A. 3 30. For acute angles P and Q, sin P = 1 13 and sin Q = 3 5. Show that the eact value of sin(p + Q) is 63 65. 3 hsn.uk.net Page 14 Questions marked c SQA
PSf 31. Find the eact value of sin θ + sin(θ + 10 ) + cos(θ + 150 ). 3 3. If is an acute angle such that tan = 3 4, show that the eact value of sin( + 30 ) is 4 3 + 3. 3 10 hsn.uk.net Page 15 Questions marked c SQA
PSf 33. 34. A and B are acute angles such that tan A = 4 3 and tan B = 1 5. Find the eact value of (a) sin A (b) cos A 1 (c) sin(a + B). hsn.uk.net Page 16 Questions marked c SQA
PSf 35. Functions f () = sin, g() = cos and h() = + π 4 set of real numbers. (a) Find epressions for: (i) f (h()); are defined on a suitable (ii) g(h()). (b) (i) Show that f (h()) = 1 sin + 1 cos. (ii) Find a similar epression for g(h()) and hence solve the equation f (h()) g(h()) = 1 for 0 π. 5 Part Marks Level Calc. Content Answer U C3 (a) C NC A4 (i) sin( + π 4 ), (ii) 001 P1 Q7 cos( + π 4 ) (b) 5 C NC T8, T7 (i) proof, (ii) = π 4, 3π 4 1 ic: interpret composite functions ic: interpret composite functions 3 ss: epand sin( + π 4 ) 4 ic: interpret 5 ic: substitute 6 pd: start solving process pd: process 7 1 sin( + π 4 ) cos( + π 4 ) 3 sin cos π 4 + cos sin π 4 and complete 4 g(h()) = 1 cos 1 sin 5 ( 1 sin + 1 cos ) ( 1 cos 1 sin ) 6 sin 7 = π 4, 3π 4 accept onl radians hsn.uk.net Page 17 Questions marked c SQA
PSf 36. n the coordinate diagram shown, A is the point (6, 8) and B is the point (1, 5). PSfrag Angle A(6, 8) AC = p and angle CB = q. Find the eact value of sin(p + q). 4 p C q B(1, 5) Part Marks Level Calc. Content Answer U C3 4 C NC T9 63 65 000 P1 Q1 1 ss: know to use trig epansion sin p cos q + cos p sin q pd: process missing sides PSf 10 and 13 3 ic: interpret data 3 8 4 10 1 13 + 10 6 5 13 pd: process 4 16 130 1 37. In triangle ABC, show that the eact C value of sin(a + b) is. 4 5 1 a b A B 1 3 Part Marks Level Calc. Content Answer U C3 4 C NC T9 proof 00 P1 Q5 pd: process the missing sides ss: epand 3 pd: substitute pc: process and complete proof 1 4 1 AC = and BC = 10 stated or implied b 3 sin(a + b) = sin a cos b + cos a sin b 3 1 3 10 + 1 1 10 4 4 0 =... = 5 hsn.uk.net Page 18 Questions marked c SQA
PSf 38. hsn.uk.net Page 19 Questions marked c SQA
PSf 39. [END F WRITTEN QUESTINS] hsn.uk.net Page 0 Questions marked c SQA