P1 TRIGONOMETRY PAST YEARS QUESTIONS

1. The equation of a curve is y = 3 cos 2x. The equation of a line is x + 2y = . On the same
diagram, sketch the curve and the line for 0 ≤ x ≤ . [4]
[9709 W2009_11]
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2. Solve the equation 3 tan(2x + 15) = 4 for 0 ≤ x ≤ 180. [4]
[ANS:19.1, 109.1]
[9709 W2009_11]
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3. (i) Prove the identity (sin x + cos x)(1 sin x cos x) = sin3 x + cos3 x. [3]
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(ii) Solve the equation (sin x + cos x)(1 sin x cos x) = 9 sin3 x for 0 ≤ x ≤ 360. [3]
[ANS:26.6, 206.6]
[9709 W2009_12]
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TRIGONOMETRY PYP 1 TanWC/SunwayCollege/2024

4*. The acute angle x radians is such that tan x = k, where k is a positive constant.
Express, in terms of k,
(i) tan ( − x), [1]
1 
(ii) tan    x  , [1]
2 
(iii) sin x. [2]
1 k
[ANS:−k, , ]
k 1 k 2
[9709 S2010_11]
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5. (i) Show that the equation 3(2 sin x − cos x) = 2(sin x − 3 cos x) can be written in the
3
form tan x =  . [2]
4
(ii) Solve the equation 3(2 sin x − cos x) = 2(sin x − 3 cos x), for 0 ≤ x ≤ 360. [2]
[ANS:143.1,323.1]
[9709 S2010_12]
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6. The function f is such that f(x) = 2 sin2x − 3 cos2x for 0 ≤ x ≤ p.
(i) Express f(x) in the form a + b cos2x, stating the values of a and b. [2]
(ii) State the greatest and least values of f(x). [2]
(iii) Solve the equation f(x) + 1 = 0. [3]
[ANS:25cos2x; 3, 2 ;0.685, 2.46]
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[9709 S2010_11]

TRIGONOMETRY PYP 2 TanWC/SunwayCollege/2024

7. The function f : x  4  3 sin x is defined for the domain 0  x  2.
(i) Solve the equation f(x) = 2. [3]
(ii) Sketch the graph of y = f(x). [2]
(iii) Find the set of values of k for which the equation f(x) = k has no solution. [2]
1
The function g : x  4  3 sin x is defined for the domain   x  A .
2
(iv) State the largest value of A for which g has an inverse. [1]
(v) For this value of A, find the value of g1(3). [2]
3
[ANS:0.730, 2.41, k<1, k>7, , 2.80]
2
[9709 S2010_12]
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8. (i) Show that the equation 2 sin x tan x + 3 = 0 can be expressed as 2 cos2 x − 3 cos x − 2 = 0.
[2]
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(ii) Solve the equation 2 sin x tan x + 3 = 0 for 0 ≤ x ≤ 360. [3]
[ANS:120,240]
[9709 S2010_13]
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TRIGONOMETRY PYP 3 TanWC/SunwayCollege/2024

 sin x tan x 1
9. (i) Prove the identity  1 . [3]
1  cos x cos x
sin x tan x
(ii) Hence, solve the equation  2  0 , for 0  x  360. [3]
1  cos x
[ANS: 109.5º, 250.5º]
[9709 W2010_11]
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10. Prove the identity tan2 x − sin2 x ≡ tan2 x sin2 x. [4]
[9709 W2010_12]
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11. Solve the equation 15 sin2 x = 13 + cos x for 0 ≤ x ≤ 180. [4]
[ANS:113.6, 70.5]
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[9709 W2010_13]

TRIGONOMETRY PYP 4 TanWC/SunwayCollege/2024

12. (i) Sketch the curve y = 2 sin x for 0 ≤ x ≤ 2. [1]
(ii) By adding a suitable straight line to your sketch, determine the number of real roots
of the equation 2 sin x =  − x.
State the equation of the straight line. [3]
x
[ANS: 1  ,3]

[9709 W2010_13]
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13. (i) Show that the equation 2 tan2 sin2 = 1 can be written in the form 2 sin4 + sin2 −1 = 0.
[2]
(ii) Hence solve the equation 2 tan2 sin2 = 1 for 0 ≤  ≤ 360. [4]
[ANS: 45°, 135°,225°, 315°]
[9709 S2011_11]
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cosq 1
14. (i) Prove the identity  1 . [3]
tan q (1 sin q ) sin q
cosq
(ii) Hence, solve the equation  4 , for 0  x  360. [3]
tan q (1 sin q )
[ANS: 19.5º, 160.5º]
[9709 S2011_12]
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TRIGONOMETRY PYP 5 TanWC/SunwayCollege/2024

15.* The function f is such that f(x) = 3 − 4 cosk x, for 0 ≤ x ≤ , where k is a constant.
(i) In the case where k = 2,
(a) find the range of f, [2]
(b) find the exact solutions of the equation f(x) = 1. [3]
(ii) In the case where k = 1,
(a) sketch the graph of y = f(x), [2]
(b) state, with a reason, whether f has an inverse. [1]
1 3
[ANS:1≤f(x)≤3, ,  ]
4 4
[9709 S2011_12]
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2
 1 1  1 cosq
16. (i) Prove the identity     . [3]
 sinq tanq  1 cosq
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2
 1 1  2
(ii) Hence, solve the equation     , for 0  x  360. [3]
 sinq tanq  5
[ANS: 64.6° , 295.4°]
[9709 S2011_13]
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TRIGONOMETRY PYP 6 TanWC/SunwayCollege/2024

 1
17. (i) Sketch, on a single diagram, the graphs of y = cos 2 and y = for 0 ≤  ≤ 2. [3]
2
(ii) Write down the number of roots of the equation 2 cos 2 −1 = 0 in the interval
0 ≤  ≤ 2. [1]
(iii) Deduce the number of roots of the equation 2 cos 2 − 1 = 0 in the interval
10 ≤  ≤ 20. [1]
[ANS:4,20]
[9709 W2011_11]
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18. (i) Sketch, on the same diagram, the graphs of y = sin x and y = cos 2x for 0 ≤ x ≤ 180. [3]
(ii) Verify that x = 30 is a root of the equation sin x = cos 2x, and state the other root of
this equation for which 0 ≤ x ≤ 180. [2]
(iii) Hence state the set of values of x, for 0 ≤ x ≤ 180, for which sin x < cos 2x. [2]
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[ANS:150,0x<30 150<x180]]
[9709 W2011_12]
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TRIGONOMETRY PYP 7 TanWC/SunwayCollege/2024

 2
19. (i) Given that 3 sin2 x − 8 cos x − 7 = 0, show that, for real values of x, cos x =  . [3]
3
(ii) Hence solve the equation 3 sin2( + 70) − 8 cos( + 70) − 7 = 0
for 0≤  ≤ 180. [4]
[ANS: 61.8,158.2]
[9709 W2011_12]
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20.*(i) Prove the identity tan2 − sin2 ≡ tan2 sin2. [3]
(ii) Use this result to explain why tan  > sin  for 0 <  < 90. [1]
[9709 S2012_13]
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21*. (i) Solve the equation 2 cos2  = 3 sin , for 0    360. [4]
(ii) The smallest positive solution of the equation 2 cos2(n) = 3 sin(n), where n is a positive
integer, is 10. State the value of n and hence find the largest solution of this equation in
the interval 0    360. [3]
[ANS:30, 150, 3, 290]
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[9709 W2012_11]

TRIGONOMETRY PYP 8 TanWC/SunwayCollege/2024

 sin q cosq 1
22. (i) Show that   2 . [3]
sin q  cosq sinq  cosq sin q  cos2 q
sin q cosq
(ii) Hence solve the equation   3 , for 0º    360º. [4]
sin q  cosq sin q  cosq
[ANS: 54.7°, 125.3°, 234.7°, 305.3°]
[9709 S2013_11]
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23. It is given that a = sin  − 3 cos  and b = 3 sin  + cos , where 0 ≤  ≤ 360.
(i) Show that a2 + b2 has a constant value for all values of . [3]
(ii) Find the values of  for which 2a = b. [4]
[ANS:98.1,278.1]
[9709 S2013_12]
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24*. (i) Express the equation 2 cos2 = tan2 as a quadratic equation in cos2. [2]
(ii) Solve the equation 2 cos2 = tan2 for 0 ≤  ≤ , giving solutions in terms of . [3]
1 3
[ANS:c4+c2+1,  ,  ]
4 4
[9709 S2013_13]
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TRIGONOMETRY PYP 9 TanWC/SunwayCollege/2024

 1
( )
25. (a) Find the possible values of x for which sin 1 x 2 1   , giving your answers correct
3
to 3 decimal places. [3]
 1  1
(b) Solve the equation sin  2q     for 0    , giving  in terms of  in your
 3  2
answers. [4]
1 11
[ANS:1.366,  ,  ]
4 12
[9709 S2013_13]
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26. (i) Sketch, on the same diagram, the curves y = sin 2x and y = cos x − 1 for 0 ≤ x ≤ 2. [4]
(ii) Hence state the number of solutions, in the interval 0 ≤ x ≤ 2, of the equations
(a) 2 sin 2x + 1 = 0, [1]
(b) sin 2x − cos x + 1 = 0. [1]
[ANS:4,3]
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[9709 S2013_13]
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27. (i) Solve the equation 4 sin2x + 8 cos x – 7 = 0 for 0  x  360. [4]
1  1 
(ii) Hence find the solution of the equation 4sin 2  q   8cos  q   7  0 0  x  360. [2]
2  2 
[ANS: 60º, 300º,120]
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[9709 W2013_11]

TRIGONOMETRY PYP 10 TanWC/SunwayCollege/2024

28. Given that cos x = p, where x is an acute angle in degrees, find, in terms of p,
(i) sin x, [1]
(ii) tan x, [1]
(iii) tan(90− x). [1]

1 p 2 p
[ANS: 1 p2 , , ]
p 1 p 2
[9709 W2013_12]
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sin q 1 1
29. (i) Prove the identity   . [4]
1 cosq sin q tan q

sin q 1
(ii) Hence solve the equation   4 tanq for 0º <  < 180º. [3]
1 cosq sin q
[ANS: 26.6º, 153.4º]
[9709 S2014_11]
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30.
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The diagram shows part of the graph of y = a + b sin x. State the values of the constants
a and b. [2]
[9709_S2014_11]

TRIGONOMETRY PYP 11 TanWC/SunwayCollege/2024

31.* The reflex angle  is such that cos  = k, where 0 < k < 1.
(i) Find an expression, in terms of k, for
(a) sin , [2]
(b) tan . [1]
(ii) Explain why sin 2 is negative for 0 < k < 1. [2]
2
 1 k
[ANS:  1  k 2 , , 540<2<720]
k
[9709 S2014_12]
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tan x 1
32. (i) Prove the identity  sin x  cos x . [3]
sin x tan x  cos x

tan x  1
(ii) Hence solve the equation  3 sin x  2 cos x for 0  x  2. [3]
sin x tan x  cos x
[ANS: 0.983, 4.12]
[9709 S2014_13]
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13 sin 2 q
33. Solve the equation  cosq  2 for 0 ≤ x ≤ 180. [4]
2  cosq
C
[ANS: 30º, 150º]
[9709 W2014_11]

TRIGONOMETRY PYP 12 TanWC/SunwayCollege/2024

34. (i) Show that the equation 1 + sin x tan x = 5 cos x can be expressed as
6 cos2 x − cos x − 1 = 0. [3]
(ii) Hence solve the equation 1 + sin x tan x = 5 cos x for 0 ≤ x ≤ 180. [3]
[ANS: 60º, 109.5º]
[9709 W2014_12]
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1 
35. The function f : x  6  4 cos  x  is defined for 0 ≤ x ≤ 2.
2 
(i) Find the exact value of x for which f(x) = 4. [3]
(ii) State the range of f. [2]
(iii) Sketch the graph of y = f(x). [2]
(iv) Find an expression for f −1(x). [3]
2 6 x
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[ANS: x   ; 2 ≤ f(x) ≤ 10; 2 cos 1  ]
3  4 
[9709 W2014_12]
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36. (i) Show that sin 4 q  cos4 q  2 sin 2 q  1 . [3]
1
(ii) Hence solve the equation sin 4 q  cos4 q  for 0    360. [4]
2
C
[ANS:60,120, 240, 300]
[9709 W2014_13]

TRIGONOMETRY PYP 13 TanWC/SunwayCollege/2024

 1 
37. The function f : x → 5  3cos x  is defined for 0 ≤ x ≤ 2.
2 
(i) Solve the equation f(x) = 7, giving your answer correct to 2 decimal places. [3]
(ii) Sketch the graph of y = f(x). [2]
(iii) Explain why f has an inverse. [1]
(iv) Obtain an expression for f 1(x). [3]
 x5
[ANS:1.68, x  2cos -1  ]
 3 
[9709 S2015_11]
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38.* A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The
wheel turns in such a way that the height, h m, of a passenger above the ground is given by
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the formula h = 60(1 − cos kt). In this formula, k is a constant, t is the time in minutes that has
elapsed since the passenger started the ride at ground level and kt is measured in radians.
(i) Find the greatest height of the passenger above the ground. [1]
One complete revolution of the wheel takes 30 minutes.
1
(ii) Show that k   . [2]
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15
(iii) Find the time for which the passenger is above a height of 90 m. [3]
[ANS:120,10 or 20 min]
[9709 S2015_12]
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TRIGONOMETRY PYP 14 TanWC/SunwayCollege/2024

 2
 1 1  1  cos x
39. (i) Prove the identity     . [4]
 sin x tan x  1  cos x
2
 1 1  2
(ii) Hence solve the equation     for 0 ≤ x ≤ 2. [3]
 sin x tan x  5
[ANS: 1.13 or 5.16]
[9709 W2015_12]
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1
40. (a) Show that the equation  3sin q tan q  4  0 can be expressed as
cos q

3 cos2 q − 4 cos q − 4 = 0,
1
and hence solve the equation  3sin q tan q  4  0 for 0 ≤  ≤ 360. [6]
cos q
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(b)
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The diagram shows part of the graph of y = a cos x − b, where a and b are constants.
The graph crosses the x-axis at the point C (cos−1 c, 0) and the y-axis at the point D(0, d).
Find c and d in terms of a and b. [2]
C
b
[ANS: 131.8 or 228.2; c  ,d=ab]
a
[9709 W2015_13]

TRIGONOMETRY PYP 15 TanWC/SunwayCollege/2024

 1
41. (a) Solve the equation sin 1 ( 3x )    , giving the solution in an exact form. [2]
3
(b) Solve, by factorising, the equation 2 cos q sin q  2 cos q  sin q  1  0 for 0 ≤  ≤ . [4]

3  
[ANS: x   ; , ]
6 3 2
[9709 M2016_12]
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1 1
42*. The function f is defined by f : x → 4 sin x − 1 for    x   .
2 2
(i) State the range of f. [2]
(ii) Find the coordinates of the points at which the curve y = f(x) intersects the coordinate
axes. [3]
(iii) Sketch the graph of y = f(x). [2]
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(iv) Obtain an expression for f −1(x), stating both the domain and range of f −1. [4]
 x 1 1 1
[ANS: 5  f ( x )  3; ( 0.253, 0 ) , ( 0, 1) ;sin 1  1
;  5  x  3 , -   f   ]
 4  2 2
[9709 S2016_11]
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TRIGONOMETRY PYP 16 TanWC/SunwayCollege/2024

43*.A function f is defined by f : x → 5 − 2 sin 2x for 0 ≤ x ≤ .
(i) Find the range of f. [2]
(ii) Sketch the graph of y = f(x). [2]
(iii) Solve the equation f(x) = 6, giving answers in terms of . [3]
The function g is defined by g : x → 5 − 2 sin 2x for 0 ≤ x ≤ k, where k is a constant.
(iv) State the largest value of k for which g has an inverse. [1]
−1
(v) For this value of k, find an expression for g (x). [3]
[ANS:3f(x)7;
7 11  1
 ,  ; ; g 1 (x)  sin 1
(5  x ) ]
12 12 4 2 2
[9709 W2016_12]
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44.
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The diagram shows the graphs of y = tan x and y = cos x for 0 ≤ x ≤ 2. The graphs intersect
at points A and B.
(i) Find by calculation the x-coordinate of A. [4]
C
(ii) Find by calculation the coordinates of B. [3]
[ANS:0.666; (2.48,0.786)]
[9709 M2017_12]

TRIGONOMETRY PYP 17 TanWC/SunwayCollege/2024

 1  cosq sin q 2
45. (i) Prove the identity   . [3]
sin q 1  cosq sin q
1  cosq sin q 3
(ii) Hence solve the equation   for 0 ≤ q ≤ 360. [3]
sin q 1  cosq cos q
[ANS: θ = 33.7° or 213.7°]
[9709 S2017_11]
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46. The equation of a curve is y = 2 cos x.
(i) Sketch the graph of y = 2 cos x for − ≤ x ≤ , stating the coordinates of the point of
intersection with the y-axis. [2]
1
Points P and Q lie on the curve and have x-coordinates of  and  respectively.
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3
(ii) Find the length of PQ correct to 1 decimal place. [2]
The line through P and Q meets the x-axis at H(h, 0) and the y-axis at K(0, k).
5
(iii) Show that h   and find the value of k. [3]
9
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5
[ANS: ; 3.7; k  ]
2
[9709 S2017_11]
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TRIGONOMETRY PYP 18 TanWC/SunwayCollege/2024

 2
 1  1  sin q
47. (i) Prove the identity   tanq   . [3]
 cos q  1+sin q
2
 1  1
(ii) Hence solve the equation   tanq   for 0 ≤ q ≤ 360. [3]
 cos q  2
[ANS: θ = 19.5º or 160.5°]
[9709 S2017_12]
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1  1 1
48. The function f is defined by f ( x )  3tan  x   2 , for    x   .
2  2 2
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(i) Solve the equation f(x) + 4 = 0, giving your answer correct to 1 decimal place. [3]
(ii) Find an expression for f −1(x) and find the domain of f −1. [5]
(iii) Sketch, on the same diagram, the graphs of y = f(x) and y = f −1(x). [3]
 x2
[ANS:x=1.2; f 1 ( x )  2tan 1   , 5 ≤ x ≤ 1 ; ]
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 3 
[9709 S2017_12]
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TRIGONOMETRY PYP 19 TanWC/SunwayCollege/2024

 2sinq  cosq
49. (i) Show that the equation  2tanq may be expressed as cos2  = 2 sin2 . [3]
sin q  cosq

2sinq  cosq
(ii) Hence solve the equation  2tanq for 0 ≤ q ≤ 180. [3]
sin q  cosq
[ANS: θ = 35.3° or 144.7°]
[9709 S2017_13]
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50. (a)
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The diagram shows part of the graph of y = a + b sin x. Find the values of the constants
a and b. [2]
(b) (i) Show that the equation
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(sin  + 2 cos )(1 + sin  − cos ) = sin (1 + cos )
may be expressed as 3 cos2  − 2 cos  − 1 = 0. [3]
(ii) Hence solve the equation
(sin  + 2 cos )(1 + sin  − cos ) = sin (1 + cos )
for −180 ≤ 1 ≤ 180. [4]
C
[ANS: a = −2, b = 3; θ = 0° or 109.5° or −109.5°]
[9709 W2017_11]

TRIGONOMETRY PYP 20 TanWC/SunwayCollege/2024

51. (i) Show that the equation cos 2x(tan22x + 3) + 3 = 0 can be expressed as
2 cos22x + 3 cos 2x + 1 = 0. [3]

(ii) Hence solve the equation cos 2x(tan22x + 3) + 3 = 0 for 0 ≤ x ≤ 180. [4]
[ANS:θ = 60° or 90° or 120°]
[9709 W2017_12]
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1 
52. (a) The function f, defined by f : x .→ a + b sin x for xℝ, is such that f     4 and
6 
1 
f     3.
2 
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(i) Find the values of the constants a and b. [3]

(ii) Evaluate ff(0). [2]

(b) The function g is defined by g : x → c + d sin x for xℝ. The range of g is given by
−4 ≤ g(x) ≤ 10.
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Find the values of the constants c and d. [3]
[ANS: a=5, b= −2; ff(0)=6.92 ; c=3, d=7]
[9709 W2017_12]
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TRIGONOMETRY PYP 21 TanWC/SunwayCollege/2024

 cosq  4
53. (i) Show that the equation  5sinq  5  0 may be expressed as
sin q  1
5 cos2 q  cos q  4 = 0. [3]
cosq  4
(ii) Hence solve the equation  5sinq  5  0 for 0 ≤ q ≤ 360. [4]
sin q  1
[ANS: θ = 0, 143.1°, 216.9 or 360°]
[9709 W2017_13]
TA
5  2 tan x
54. (a) Express the equation  1  tan x as a quadratic equation in tan x and hence
3  2 tan x
solve the equation for 0 ≤ x ≤ . [4]
N
(b)
W
The diagram shows part of the graph of y = k sin ( + ), where k and  are constants
and 0 <  < 180. Find the value of  and the value of k. [2]
[ANS:0.464, 2.03; 30 ; 4]
C
[9709 M2018_12]

TRIGONOMETRY PYP 22 TanWC/SunwayCollege/2024

55. (i) Prove the identity (sin  + cos )(1 − sin  cos )  sin3  + cos3 . [3]
(ii) Hence solve the equation (sin  + cos )(1 − sin  cos ) = 3 cos3  for 0 ≤  ≤ 360. [3]
[ANS: 51.6, 231.6]
[9709 S2018_11]
TA
1 
56. The function f is such that f(x) = a + b cos x for 0 ≤ x ≤ 20. It is given that f     5 and
3 
f() = 11.
(i) Find the values of the constants a and b. [3]
(ii) Find the set of values of k for which the equation f(x) = k has no solution. [3]
[ANS: a = 7, b = – 4; k < 3 , k > 11]
[9709 S2018_12]
N
W
57. (i) Solve the equation 2 cos x + 3 sin x = 0, for 0 ≤ x ≤ 360. [3]

(ii) Sketch, on the same diagram, the graphs of y = 2 cos x and y = −3 sin x for 0 ≤ x ≤ 360.
[3]
(iii) Use your answers to parts (i) and (ii) to find the set of values of x for 0 ≤ x ≤ 360 for
which 2 cos x + 3 sin x > 0. [2]
[ANS: 146.3, 326,3; ; x < 146.3º, x > 326.3º]
C
[9709 S2018_12]

TRIGONOMETRY PYP 23 TanWC/SunwayCollege/2024

 tan 2q  1
58. (a) (i) Express 2
in the form a sin2 q + b, where a and b are constants to be found. [3]
tan q  1
(ii) Hence, or otherwise, and showing all necessary working, solve the equation
tan 2q  1 1

tan 2q  1 4
for −90 ≤ 1 ≤ 0. [2]

(b)
TA
The diagram shows the graphs of y = sin x and y = 2 cos x for − ≤ x ≤ . The graphs intersect
at the points A and B.
(i) Find the x-coordinate of A. [2]
N
(ii) Find the y-coordinate of B. [2]
[ANS: 2sin2q 1; 52.2; x=1,11, y=0.895]
[9709 S2018_13]
W
C

TRIGONOMETRY PYP 24 TanWC/SunwayCollege/2024

 cos q  4 4 sin q
59. (i) Show that the equation  0
sin q 5 cos q  2
may be expressed as 9cos2 − 22 cos + 4 = 0. [3]

cos q  4 4 sin q
(ii) Hence solve the equation  0
sin q 5 cos q  2
for 0 ≤  ≤ 360. [3]
[ANS: 78.6, 281.4
[9709 W2018_11]
TA
60.
N
W
The diagram shows a triangle ABC in which BC = 20 cm and angle ABC = 90. The
perpendicular from B to AC meets AC at D and AD = 9 cm. Angle BCA = .
(i) By expressing the length of BD in terms of  in each of the triangles ABD and DBC,
show that 20 sin2  = 9 cos . [4]
(ii) Hence, showing all necessary working, calculate . [3]
C
[ANS: 36.9
[9709 W2018_12]

TRIGONOMETRY PYP 25 TanWC/SunwayCollege/2024

 tan q  1 tan q  1 2(tan q  cos q )
61. (i) Show that   . [3]
1  cosq 1  cos q sin 2 q
tan q  1 tan q  1
(ii) Hence, showing all necessary working, solve the equation  0
1  cosq 1  cos q
for 0 <  < 90. [4]
[ANS:38.2]
[9709 W2018_13]
TA
62. (a) Solve the equation 3 sin2 2 + 8 cos 2 = 0 for 0 ≤  ≤ 180. [5]

(b)
N
W
The diagram shows part of the graph of y = a + tan bx, where x is measured in radians and
 1 
( )
a and b are constants. The curve intersects the x-axis at    , 0  and the y-axis at 0, 3 .
 6 
Find the values of a and b. [3]
C
[ANS: =54.7 or 125.3; a  3 , b=2]
[9709 M2019_12]

TRIGONOMETRY PYP 26 TanWC/SunwayCollege/2024

 2
 1  1  sin x
63. (i) Prove the identity   tan x   . [4]
 cos x  1  sin x
2
 1  1
(ii) Hence solve the equation   tan 2 x   for 0  x  . [3]
 cos 2 x  3
 5
[ANS: , ]
12 12
[9709 S2019_11]
TA
64. Angle x is such that sin x = a + b and cos x = a − b, where a and b are constants.
(i) Show that a2 + b2 has a constant value for all values of x. [3]
(ii) In the case where tan x = 2, express a in terms of b. [2]
1
[ANS: a 2  b 2  ; a=3b]
2
[9709 S2019_12]
N
W
3x
65. The equation of a curve is y = 3 cos 2x and the equation of a line is 2 y  5.

(i) State the smallest and largest values of y for both the curve and the line for 0 ≤ x ≤ 2. [3]
3x
(ii) Sketch, on the same diagram, the graphs of y = 3 cos 2x and 2 y   5 for 0 ≤ x ≤ 2. [3]
C

3x
(iii) State the number of solutions of the equation 6cos 2 x  5  for 0 ≤ x ≤ 2. [1]

[ANS: 3, 3; 4]
[9709 S2019_12]

TRIGONOMETRY PYP 27 TanWC/SunwayCollege/2024

66.

The function f : x → p sin2 2x + q is defined for 0 ≤ x ≤ , where p and q are positive constants.
The diagram shows the graph of y = f(x).
(i) In terms of p and q, state the range of f. [2]
TA
(ii) State the number of solutions of the following equations.
(a) f(x) = p + q [1]
(b) f(x) = q [1]
1
(c) f(x) = p + q [1]
2
(iii) For the case where p = 3 and q = 2, solve the equation f(x) = 4, showing all necessary
working. [5]
[ANS: q  f(x)  p + q ; 2,3,4; 0.478, 1.09, 2.05, 2.66]
[9709 S2019_13]
N
W
1 2
67. (i) Given that 4 tan x  3cos x   0 , show, without using a calculator, that sin x   .
cos x 3
[3]
(ii) Hence, showing all necessary working, solve the equation
1
4 tan ( 2 x  20 )  3cos ( 2 x  20 )  0
cos ( 2 x  20 )
C
for 0  x  180. [4]
[ANS: 120.9º, 169.1º]
[9709 W2019_11]

TRIGONOMETRY PYP 28 TanWC/SunwayCollege/2024

68. (a) Given that x > 0, find the two smallest values of x, in radians, for which 3 tan(2x + 1) = 1.
Show all necessary working. [4]

(b) The function f : x → 3 cos2x − 2 sin2x is defined for 0 ≤ x ≤ .

(i) Express f(x) in the form a cos2 x + b, where a and b are constants. [1]
(ii) Find the range of f. [2]
[ANS: 1.23, 2.80; 5 cos2 x – 2; 2f3]
[9709 W2019_12]
TA

69. (i) Show that the equation 3 cos4  + 4 sin2  − 3 = 0 can be expressed as 3x2 − 4x + 1 = 0,
N
where x = cos2 . [2]
(ii) Hence solve the equation 3 cos4  + 4 sin2  − 3 = 0 for 0 ≤  ≤ 180. [5]
[ANS: 0º, 180º, 54.7º, 125.3º]
[9709 W2019_13]
W
C

TRIGONOMETRY PYP 29 TanWC/SunwayCollege/2024

70.

3 1
The diagram shows the graph of y = f(x), where f(x) = cos 2 x  for 0 ≤ x ≤ π.
2 2
(a) State the range of f. [2]
TA
A function g is such that g(x) = f(x) + k, where k is a positive constant. The x-axis is a
tangent to the curve y = g(x).
(b) State the value of k and hence describe fully the transformation that maps the curve
y = f(x) on to y = g(x). [2]
(c) State the equation of the curve which is the reflection of y = f(x) in the x-axis. Give your
answer in the form y = a cos 2x + b, where a and b are constants. [1]
0 3 1
[ANS: 1≤f(x)≤2; k=1, translation   ; y   cos 2 x  ]
1 2 2
[9709 S2020_11]
N
W
C

TRIGONOMETRY PYP 30 TanWC/SunwayCollege/2024

71. Functions f and g are such that
f(x) = 2 − 3 sin 2x for 0 ≤ x ≤ π,
g(x) = −2f(x) for 0 ≤ x ≤ π.
(a) State the ranges of f and g. [3]
The diagram below shows the graph of y = f(x).
TA
N
(b) Sketch, on this diagram, the graph of y = g(x). [2]
The function h is such that
h(x) = g(x + π) for π ≤ x ≤ 0.
(c) Describe fully a sequence of transformations that maps the curve y = f(x) on to y = h(x). [3]
W
[ANS: 1≤f(x)≤5; 10≤g(x) ≤21; ; Reflection in x-axis, Stretch by factor 2 in
 0 
y-direction, translation   ]
 π 
[9709 S2020_12]
C

TRIGONOMETRY PYP 31 TanWC/SunwayCollege/2024

72.
TA
In the diagram, the lower curve has equation y = cos q. The upper curve shows the result of
applying a combination of transformations to y = cos q.
Find, in terms of a cosine function, the equation of the upper curve. [3]
1
[ANS: y  3  2cos q ]
2
[9709 W2020_11]

73. A curve has equation y = 3 cos 2x + 2 for 0 ≤ x ≤ π.

(a) State the greatest and least values of y. [2]
N
(b) Sketch the graph of y = 3 cos 2x + 2 for 0 ≤ x ≤ π. [2]
(c) By considering the straight line y = kx, where k is a constant, state the number of solutions
of the equation 3 cos 2x + 2 = kx for 0 ≤ x ≤ π in each of the following cases.
(i) k = −3 [1]
W
(ii) k = 1 [1]
(iii) k = 3 [1]
Functions f, g and h are defined for xℝ by
f(x) = 3 cos 2x + 2,
g(x) = f(2x) + 4,
 1 
. h ( x )  2f  x  π 
C
 2 
(d) Describe fully a sequence of transformations that maps the graph of y = f(x) on to
y = g(x). [2]
(e) Describe fully a sequence of transformations that maps the graph of y = f(x) on to
y = h(x). [2]
1  0
[ANS:5, 1; ; 0, 2, 1; Stretch by factor in x-direction, translation  ;
2  4
 π

translation  2  , Stretch by factor 2 in y-direction]
 
 0 
[9709 W2020_12]
TRIGONOMETRY PYP 32 TanWC/SunwayCollege/2024
 tan q  2sin q
74. Solve the equation  3 for 0 < q < 180. [4]
tan q  2sin q
[ANS:75.5]
[9709 M2021_12]

75.
TA
N
The diagram shows part of the graph of y = a tan (x − b) + c.
Given that 0 < b < π, state the values of the constants a, b and c. [3]

[ANS: a=2, b  , c=1]
W
4
[9709 S2021_11]

1  2sin 2 q
76. (a) Prove the identity  1  tan 2 q . [2]
1  sin 2 q
1  2sin 2 q
(b) Hence solve the equation  2 tan 4 q for 0 ≤  ≤ 180. [3]
1  sin 2 q
C
[ANS:35.3, 144.7]
[9709 S2021_11]

TRIGONOMETRY PYP 33 TanWC/SunwayCollege/2024

 1  sin x 1  sin x 4 tan x
77. (a) Prove the identity   . [4]
1  sin x 1  sin x cos x

1  sin x 1  sin x 1
(b) Hence solve the equation   8 tan x for 0  x   . [3]
1  sin x 1  sin x 2

[ANS: x  0, ]
3
[9709 S2021_12]
TA
78. (a) The graph of y = f(x) is transformed to the graph of y = 2f(x − 1).
Describe fully the two single transformations which have been combined to give the
resulting transformation. [3]
1
N
(b) The curve y = sin 2x − 5x is reflected in the y-axis and then stretched by scale factor
3
in the x-direction.
Write down the equation of the transformed curve. [2]
1
[ANS: Translation   , Stretch, factor 2 in y-direction; sin 6x+15x or sin(6x)+15x]
0
W
[9709 S2021_12]
C

TRIGONOMETRY PYP 34 TanWC/SunwayCollege/2024

79. (a) Show that the equation
tan x  sin x
 k,
tan x  sin x
where k is a constant, may be expressed as
1  cos x
 k. [2]
1  cos x
(b) Hence express cos x in terms of k. [2]
tan x  sin x
(c) Hence solve the equation  4 for  < x < . [2]
tan x  sin x
k 1
[ANS: cos x  ; 0.927]
k 1
TA
[9709 S2021_13]

80. Solve, by factorising, the equation

N
6 cos  tan  − 3 cos  + 4 tan  − 2 = 0,
for 0 ≤  ≤ 180. [4]
[ANS: 26.6º, 131.8º]
[9709 W2021_11]
W
3
81. Solve the equation 2cos q  7  for 90 ≤ q ≤ 90. [4]
C
cos q
[ANS: 60º, 60º]
[9709 W2021_12]

TRIGONOMETRY PYP 35 TanWC/SunwayCollege/2024

82.
TA
The diagram shows part of the graph of y = a cos (bx) + c.
(a) Find the values of the positive integers a, b and c. [3]
(b) For these values of a, b and c, use the given diagram to determine the number of
solutions in the interval 0 ≤ x ≤ 2π for each of the following equations.
6
(i) a cos ( bx )  c  x [1]

N
6
(ii) a cos ( bx )  c  6  x [1]

[ANS: a=5, b=2, c=3; 3; 2]
[9709 W2021_11]
W
83. (a) Show that the equation
tan x  cos x
 k,
tan x  cos x
where k is a constant, may be expressed as
( k  1) sin 2 x  ( k  1) sin x  ( k  1)  0. [4]

tan x  cos x
(b) Hence solve the equation  4 for 0 ≤ x ≤ 360. [4]
C
tan x  cos x
[ANS:48.1, 131.9]
[9709 W2021_13]

TRIGONOMETRY PYP 36 TanWC/SunwayCollege/2024

 2
84 (a) Solve the equation 6 y  7  0. [4]
y
2
(b) Hence solve the equation 6 tan x   7  0 for 0 ≤ x ≤ 360. [3]
tan x
1 4
[ANS: y  , y  ; 14, 24, 194, 204 ]
4 9
[9709 S2022_13]
TA
85*. (a) The curve y = sin x is transformed to the curve y  4sin ( 1
2 x  30 .
)
Describe fully a sequence of transformations that have been combined, making clear the
order in which the transformations are applied. [5]
N
(b) Find the exact solutions of the equation 4sin ( 1
2 x  30  2 2 for 0 ≤ x ≤ 360.
) [3]

 30 
[ANS: Translation   followed by stretch, factor 2 in x-direction OR Stretch, factor 2 in
 0 
 60 
x-direction followed by translation   . Lastly any sequence stretch, factor 4 in y-direction]
W
 0 
[9709 S2022_11]
C

TRIGONOMETRY PYP 37 TanWC/SunwayCollege/2024

86.

The diagram shows part of the curve with equation y = p sin(q) + r, where p, q and r are
TA
constants.
(a) State the value of p. [1]
(b) State the value of q. [1]
(c) State the value of r. [1]
1
[ANS:p = 3; q  ; r = 2]
2
[9709 S2022_13]
N
85. (a) Find the set of values of k for which the equation 8x2 + kx + 2 = 0 has no real roots. [2]
(b) Solve the equation 8 cos2  − 10 cos  + 2 = 0 for 0 ≤ 1 ≤ 180. [3]
W
[ANS: −8 < k < 8 or 8 < k ]
[9709 W2022_12]
C

TRIGONOMETRY PYP 38 TanWC/SunwayCollege/2024

86.

[ANS: 51.8, 308.2 ]

[9709 M2023_12]
TA

87.
N
[ANS: 104.5 ]
[9709 S2023_11]
W
C

TRIGONOMETRY PYP 39 TanWC/SunwayCollege/2024

88.
TA
[ANS: 0, 16  , 12  , 65  ]
[9709 S2023_12]
N
W
C

TRIGONOMETRY PYP 40 TanWC/SunwayCollege/2024

89.

[ANS: 54.7  ,125.3 ]

[9709 S2023_13]
TA

90.
N
W
[ANS: 138.6 , 221.4 ]
[9709 W2023_11]
C

TRIGONOMETRY PYP 41 TanWC/SunwayCollege/2024

91.

3
[ANS:  ]
4
[9709 W2023_12]
TA

92.
N
W
[ANS: 60 , 72 ,144 ]
[9709 W2023_12]
C

TRIGONOMETRY PYP 42 TanWC/SunwayCollege/2024

93.

[ANS:1.23, 2.09, 4.19, 5.05]

[9709 W2023_12]
TA
N
94.

[ANS: 0, 32.3, 32.3]

W
[9709 M2024_12]
C

TRIGONOMETRY PYP 43 TanWC/SunwayCollege/2024

95.

[ANS: 120, 240]

[9709 S2024_11]
TA
N
96.
W
[ANS: 228.6, 311.4]
[9709 S2024_12]
C

TRIGONOMETRY PYP 44 TanWC/SunwayCollege/2024

97.
TA

[ANS: ( 53  , 0 ) , ( 196  , k ) ; t  16 ]
N
[9709 S2024_13]
W
C

TRIGONOMETRY PYP 45 TanWC/SunwayCollege/2024

98.

( )
[ANS:  5sin 2 q  7sin q  6  0 ; 18.4, 71.6]
[9709 S2024_13]
TA

99.
N
a2
[ANS: 2
 3a 1  a 2 ; 195.5, 344.5]
1 a
W
[9709 W2024_11]
C

TRIGONOMETRY PYP 46 TanWC/SunwayCollege/2024

100.
TA
N
[ANS: a  4, b  2, c  3 ; 5; 1]
[9709 W2024_12]
W
101.


[ANS: x   ; 5; 1]
C
6
[9709 W2024_13]

TRIGONOMETRY PYP 47 TanWC/SunwayCollege/2024