Trigonometric Functions, Identities and Equations · CAPE Pure Mathematics Unit 1

206 past-paper questions on Trigonometric Functions, Identities and Equations, part of Trigonometry, Geometry and Vectors, from every CAPE Pure Mathematics Unit 1 paper on Quelpr.

1. Q191 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The function \sin\left(x + \frac{\pi}{2}\right) can be simplified to
2. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1Which of the following sketches BEST represents the curve \[ y = \cos\frac{1}{2}x, \quad (0 \le x \le 2\pi)? \]
3. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1\sin(30^\circ - A) is equal to
4. Q221 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1If \beta is an acute angle and \cos\beta = \frac{5}{13}, then \sec\beta =
5. Q231 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 12\sin\theta\cos\phi is equivalent to
6. Q241 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1If 2\cos\theta + 9\sin\theta = r\cos(\theta - \alpha), where r > 0 and 0 < \alpha < \frac{\pi}{2}, then the maximum value of the expression is
7. Q251 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1Which of the following equations BEST represents the graph shown above?
8. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1The value of \sin\left(\frac{\pi}{2} - p\right) is
9. Q181 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1If \beta is an acute angle and \cos\beta = \frac{5}{13}, then \sec\beta =
10. Q191 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Which of the following sketches BEST illustrates the curve y = \cos x?
11. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1The expression \sin 6A + \sin 4A may be written as
12. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1If 2\cos\theta + 9\sin\theta \equiv r\cos(\theta - \alpha) where r > 0 and 0 < \alpha < \frac{\pi}{2}, then the maximum value of the expression is
13. Q221 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Which of the following equations BEST represents the graph shown above?
14. 3(a)6 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Prove that \frac{\tan\theta \sin\theta}{1 - \cos\theta} = 1 + \frac{1}{\cos\theta}.
15. 3(b)6 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Solve the equation \tan^2\theta - 2\tan\theta = 3 for 0 \le \theta \le 2\pi.
16. 3(c)(i)5 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Show that 4\cos\theta + 3\sin\theta = 5\sin(\theta + 0.927^c).
17. 3(c)(ii)4 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Hence, or otherwise, solve the equation 4\cos\theta + 3\sin\theta = 0.
18. 3(d)4 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Given that \sin A = \frac{1}{3} and A is obtuse, calculate the value of \cos A without using a calculator.
19. 7(a)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Express f(theta) = sqrt(2) cos theta - sin theta in the form R cos(theta + alpha).
20. 7(b)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Hence, find the minimum value of f(theta), where 0 <= theta <= 2pi.
21. 7(c)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Determine the value of theta, 0 <= theta <= 2pi, at which the minimum value of f(theta) occurs.
22. 7(a)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the EXACT length of AC.
23. 7(b)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the EXACT length of AB.
24. 8(a)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Solve the equation 4 cos^2 theta - 4 sin theta - 1 = 0 for 0 <= theta <= pi.
25. 8(b)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Show that (1 - cos 2x)/(1 + cos 2x) = tan^2 x.
26. Q161 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The graph of y = \sin 2x is
27. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The value of \sin\left(\frac{\pi}{2} + p\right) is
28. Q181 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1\frac{1}{\csc^2 x} \equiv
29. Q191 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1If P = (2\sin^2\theta + 2\cos^2\theta)(\sec^2\theta - \tan^2\theta) then P is equal to
30. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 11 + \cos^4 A - \sin^4 A =
31. Q221 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The expression \cot x + \tan x can be written as
32. Q281 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The distance (d) metres of a reciprocating arm of a shaping machine from its starting position can be modelled by the equation d = 12\cos\theta + 5\sin\theta. The MAXIMUM distance, in metres, from the starting point…
33. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1For n \in \mathbb{Z}, the general solution of \sqrt{3}\sin\theta - \cos\theta = 0 is \theta =
34. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1\frac{\sin\theta (1 - \sin^2\theta)}{\cos\theta (1 - \cos^2\theta)} =
35. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1\sin(\alpha + 45^\circ) is equal to
36. Q231 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1The minimum and maximum values of \frac{1}{2 + \sin\theta} respectively are
37. Q241 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1The general solution for \sin 2\theta = \sin\frac{\pi}{6} is
38. Q271 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1\frac{\sin\theta - \sin\alpha}{\cos\theta + \cos\alpha} is identical to
39. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The value of \sin\left(\frac{\pi}{2} + p\right) is
40. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1If \beta is an acute angle and \sin\beta = \frac{12}{13}, then \sec\beta =
41. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1\frac{1}{\text{cosec}^2 x} \equiv
42. Q221 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The expression \cot x + \tan x may be written as
43. Q231 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1What value of \theta, 0 \le \theta \le \pi, satisfies the equation 2\cos^2\theta + 3\cos\theta - 2 = 0?
44. Q251 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The expression \sin 6A + \sin 4A may be written as
45. Q281 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1Which of the following sketches BEST represents the curve y = \cos\frac{1}{2}x, (0 \le x \le 2\pi)?
46. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1Which of the following equations BEST represents the graph shown above?
47. Q301 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1If 2\cos\theta + 9\sin\theta \equiv r\cos(\theta - \alpha) where r > 0 and 0 < \alpha < \frac{\pi}{2}, then the maximum value of the expression is
48. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1\frac{\sin\theta(1 - \sin^2\theta)}{\cos\theta(1 - \cos^2\theta)} =
49. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1Which of the following sketches BEST represents the curve y = \cos\frac{1}{2}x, (0 \le x \le 2\pi)?
50. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1\sin(\alpha + 45^\circ) is equal to
51. Q251 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1Given that \alpha is an acute angle and \tan\alpha = \frac{3}{4}, then \sin(90^\circ - \alpha) =
52. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1\sin(30^\circ - A) is equal to
53. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The distance, d metres, of an arm of a shaping machine from its starting position can be modelled by the equation d = 12\cos\theta + 5\sin\theta. The MAXIMUM distance, in metres, from the starting point is
54. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1\sin(30^\circ - A) is equal to
55. Q181 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 12\sin\theta\cos\phi is equivalent to
56. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1If \beta is an acute angle and \cos\beta = \frac{5}{13}, then \sec\beta =
57. Q221 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1The value of \sin\left(\frac{\pi}{2} + p\right) is
58. Q231 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1What value of \theta, 0 \le \theta \le \pi, satisfies the equation 2\cos^2\theta + 3\cos\theta - 2 = 0?
59. Q251 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1The expression \sin 6A + \sin 4A may be written as
60. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 11 + \cos^4 A - \sin^4 A =
61. Q281 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1The general solution for \sin 2\theta = \sin\frac{\pi}{6} is
62. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1The value of \cos\left(\frac{\pi}{2} - p\right) is
63. Q191 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1The expression \sin 6\theta + \sin 4\theta may be expressed as
64. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1Which of the following sketches BEST represents the curve y = \cos\frac{1}{2}x, \quad (0 \le x \le 2\pi)?
65. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1What value of \theta, 0 \le \theta \le \pi, satisfies the equation 2\cos^2\theta + 3\cos\theta - 2 = 0?
66. Q241 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1If P = (2\sin^2\theta + 2\cos^2\theta)(\sec^2\theta - \tan^2\theta), then P is equal to
67. Q251 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1The expression \sin\left(\alpha + \frac{\pi}{4}\right) is equivalent to
68. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1The distance, d metres, of an arm of a shaping machine from its starting position can be modelled by the equation d = 12\cos\theta + 5\sin\theta. The MAXIMUM distance, in metres, from the starting point is
69. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1\sin(30^\circ - A) is equal to
70. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1If \beta is an acute angle and \cos \beta = \frac{5}{13}, then \sec \beta =
71. Q231 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1What value of \theta, 0 \le \theta \le \pi, satisfies the equation 2\cos^2 \theta + 3\cos \theta - 2 = 0?
72. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 11 + \cos^4 A - \sin^4 A \equiv
73. Q301 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1\frac{1}{\text{cosec}^2 x} =
74. 3(a)9 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Prove that (1 - \sin\theta)/(1 + \sin\theta) = (\sec\theta - \tan\theta)^2.
75. 3(b)9 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Solve the equation 2\cos^2 x - 3\sin x = 3 for 0 \le x \le 2\pi.
76. 3(c)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Show that \cos(\pi/2 + x) = -\sin x.
77. 3(d)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2A and B are acute angles such that \sin A = 3/5 and \cos B = 5/13. Calculate, without using tables or calculators, the EXACT value of \cos(A - B).
78. 3(a)(i)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Express 5 sin theta + 12 cos theta in the form r sin(theta + alpha), where r > 0 and 0 < alpha < 2*pi.
79. 3(a)(ii)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Hence, or otherwise, show that 5 sin theta + 12 cos theta + 7 <= 20.
80. 3(b)(i)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Show that 2 sin x + 3 cos x = 3 may be written as 13 cos^2 x - 18 cos x + 5 = 0.
81. 3(b)(ii)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Hence or otherwise, solve the equation 2 sin x + 3 cos x = 3 for -pi <= x <= pi.
82. 3(c)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Show that (2 cos^2 x - 1)^2 / (cos^4 x - sin^4 x) == 1 - 2 sin^2 x.
83. 3(a)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Show that 1 + tan² θ = sec² θ.
84. 3(b)(i)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Derive the identity for cos 2θ in terms of cos θ only.
85. 3(b)(ii)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Hence, solve cos 2θ - 3 cos θ = 1 for 0 ≤ θ ≤ 2π.
86. 1(b)(i)3 marks· Pure Mathematics · Unit 1 Q1 1(b)(i)Write down, in terms of θ and r, an expression for the area of S₁.
87. 1(b)(i)4 marks· Pure Mathematics · Unit 1 Q1 1(b)(i)Calculate to 3 significant figures the length of BC.
88. 1(b)(ii)4 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)Write down, in terms of θ and r, an expression for the area of S₂.
89. 1(b)(ii)4 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)Calculate to 3 significant figures the value of sin C.
90. 1(b)(iii)4 marks· Pure Mathematics · Unit 1 Q1 1(b)(iii)Given that the area of S₂ is three times the area of S₁, show that 4θ = π + 2 sin θ.
91. 1(b)(iv)5 marks· Pure Mathematics · Unit 1 Q1 1(b)(iv)Show that when r = 6, θ = π/3, the area of triangle PQR = 18√3.
92. 1(c)(i)6 marks· Pure Mathematics · Unit 1 Q1 1(c)(i)Show that the area of the shaded region is 2(π - 2√2).
93. 1(c)(ii)4 marks· Pure Mathematics · Unit 1 Q1 1(c)(ii)Using the cosine rule, show that the length of the chord AB is 4√(2 - √2).
94. 2(a)13 marks· Pure Mathematics · Unit 1 Q2 2(a)If t = tan (θ/2), express cos θ and sin θ in terms of t. Hence, find tan (θ/2) when cos θ + 2 sin θ = 11/5.
95. 2(b)4 marks· Pure Mathematics · Unit 1 Q2 2(b)Given that δ is an acute angle such that cos δ = (1/4)x, find the expression for sin 2δ in terms of x.
96. 2(b)(i)3 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)Find cos θ.
97. 2(b)(i)a)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)a)Copy and complete the following table for the function f(x) = sin x, 0≤x≤ 2π.
98. 2(b)(i)b)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)b)Sketch the graph of f.
99. 2(b)(ii)3 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)Find sin θ.
100. 2(b)(iii)3 marks· Pure Mathematics · Unit 1 Q2 2(b)(iii)Find the length of BC.
101. 2(b)(iv)3 marks· Pure Mathematics · Unit 1 Q2 2(b)(iv)Find the length of AC.
102. 2(c)(i)3 marks· Pure Mathematics · Unit 1 Q2 2(c)(i)Express cot θ and cosec θ in terms of x and y.
103. 2(d)(i)6 marks· Pure Mathematics · Unit 1 Q2 2(d)(i)Sketch, in separate diagrams, the graphs of y = sin x and y = cos x for -2π ≤ x ≤ 2π.
104. 3(a)6 marks· Pure Mathematics · Unit 1 Q3 3(a)Given that tan² x = sin² x / (1 - sin² x), show that sin x = ± tan x / √(1 + tan² x).
105. 3(a)(i)4 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Prove that (cot y - cot x)/(cot x + cot y) = sin(x - y)/sin(x + y).
106. 3(a)(i)8 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Show that (sin 2θ - cos 2θ + 1) / (cos 2θ + sin 2θ - 1) = sec 2θ + tan 2θ.
107. 3(a)(i)4 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Prove the identity tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)).
108. 3(a)(i)7 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)prove that cos 3θ = 2 cos θ [cos² θ – sin² θ – 1/2].
109. 3(a)(i)4 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Show that sin 2θ = (2 tan θ) / (1 + tan² θ)
110. 3(a)(i)4 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Show that sec²θ = cosecθ / (cosecθ - sinθ).
111. 3(a)(i)6 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Show that cos 3x = 4 cos³ x – 3 cos x.
112. 3(a)(i)5 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Express 4 sin θ + 3 cos θ in the form of R sin(θ + α).
113. 3(a)(ii)9 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)Hence, or otherwise, solve cos 6x – cos 2x = 0 for 0 < x < 2π.
114. 3(a)(ii)5 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)Using the appropriate formula, show that 1/2 [sin 6θ – sin 2θ] = (2 cos² 2θ – 1) sin 2θ.
115. 3(a)(ii)5 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)Hence, or otherwise, determine the general solution of (sin θ - cos θ + 1) / (cos θ + sin θ - 1) = 0.
116. 3(a)(ii)8 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)Hence, or otherwise, solve sin 2θ – tan θ = 0 for 0 ≤ θ ≤ 2π.
117. 3(a)(ii)5 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)Hence, solve the equation 4 sin θ + 3 cos θ = 2 for 0 ≤ θ ≤ 2π.
118. 3(a)(ii)6 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)Given that sin A = 3/5 and cos B = -1/2, where angle A is acute and angle B is obtuse, express tan (A + B) in the form a + b√3, where a and b are real numbers.
119. 3(a)(ii)5 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)Hence, or otherwise, solve the equation cosecθ / (cosecθ - sinθ) = 4/3 for 0 ≤ θ ≤ 2π.
120. 3(a)(ii)8 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)Hence, or otherwise, find the possible values for y in the trigonometric equation (cot y - cot x)/(cot x + cot y) = 1, 0 ≤ y ≤ 2π, when sin x = 1/2, 0 ≤ x ≤ π/2.
121. 3(a)(iii)5 marks· Pure Mathematics · Unit 1 Q3 3(a)(iii)Hence, or otherwise, solve sin 6θ − sin 2θ = 0 for 0 ≤ θ ≤ π/2.
122. 3(b)6 marks· Pure Mathematics · Unit 1 Q3 3(b)Solve the equation sin²θ - 2cos²θ + 3cosθ + 5 = 0 for 0 ≤ θ ≤ 4π.
123. 3(b)7 marks· Pure Mathematics · Unit 1 Q3 3(b)Calculate the value of cos(A - C).
124. 3(b)8 marks· Pure Mathematics · Unit 1 Q3 3(b)Find ALL possible values of cos θ such that 2 cot² θ + cos θ = 0.
125. 3(b)7 marks· Pure Mathematics · Unit 1 Q3 3(b)Solve, for 0° ≤ θ ≤ 180°, the equation 6 cos² θ + sin θ = 4.
126. 3(b)6 marks· Pure Mathematics · Unit 1 Q3 3(b)Show that (1 - cos 2A + sin 2A) / (1 + cos 2A + sin 2A) = tan A.
127. 3(b)7 marks· Pure Mathematics · Unit 1 Q3 3(b)Find the general solutions of the equation cos θ = 2 sin²θ – 1.
128. 3(b)(i)6 marks· Pure Mathematics · Unit 1 Q3 3(b)(i)Show that f(θ) = √3 sin θ + cos θ may be expressed as f(θ) = 2sin(θ + π/6) where 0 ≤ θ ≤ π/2.
129. 3(b)(i)6 marks· Pure Mathematics · Unit 1 Q3 3(b)(i)Express f(2θ) = 3 sin 2θ + 4 cos 2θ in the form r sin (2θ + α) where r > 0 and 0 < α < π/2.
130. 3(b)(i)4 marks· Pure Mathematics · Unit 1 Q3 3(b)(i)Express f(θ) = 3 sin 2θ + 4 cos 2θ in the form r sin (2θ + α) where r > 0 and 0 < α < π/2.
131. 3(b)(i)4 marks· Pure Mathematics · Unit 1 Q3 3(b)(i)Express f(θ) = 3 cos θ – 4 sin θ in the form r cos (θ+ α) where r > 0 and 0° ≤ α ≤ π/2.
132. 3(b)(i)5 marks· Pure Mathematics · Unit 1 Q3 3(b)(i)Express the function f(θ) = sinθ + cosθ in the form r sin(θ + α), where r > 0 and 0 ≤ α ≤ π/2.
133. 3(b)(i)3 marks· Pure Mathematics · Unit 1 Q3 3(b)(i)Prove that sin 2θ - tan θ cos 2θ = tan θ.
134. 3(b)(i)3 marks· Pure Mathematics · Unit 1 Q3 3(b)(i)Calculate sin 2A.
135. 3(b)(i)5 marks· Pure Mathematics · Unit 1 Q3 3(b)(i)Solve the following equations for 0 ≤ x ≤ π/2: 6 sin²x - cos x - 4 = 0.
136. 3(b)(ii)4 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)Hence, or otherwise, find the maximum and minimum values of 1/(7 - f(2θ)).
137. 3(b)(ii)3 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)Calculate cos(A + B).
138. 3(b)(ii)5 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)Hence, find the maximum value of f and the smallest non-negative value of θ at which it occurs.
139. 3(b)(ii)2 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)Express tan θ in terms of sin 2θ and cos 2θ.
140. 3(b)(ii)1 mark· Pure Mathematics · Unit 1 Q3 3(b)(ii)Hence, find
141. 3(b)(ii)5 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)Solve the following equations for 0 ≤ x ≤ π/2: √3 sin x + cos x = 2.
142. 3(b)(ii)a)5 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)a)solve the equation f(θ) = √2 for 0 ≤ θ ≤ 2π
143. 3(b)(ii)a)4 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)a)Hence, or otherwise, determine the value of θ, between 0 and 2π radians, at which f(θ) is a minimum
144. 3(b)(ii)a)2 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)a)the maximum value of f(θ)
145. 3(b)(ii)b)3 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)b)determine the maximum value of f and the smallest positive value of θ for which it occurs.
146. 3(b)(ii)b)5 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)b)the minimum and maximum values of 1/(7-f(θ))
147. 3(b)(ii)b)2 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)b)the minimum value of 1 / (8+f(θ))
148. 3(b)(iii)4 marks· Pure Mathematics · Unit 1 Q3 3(b)(iii)Hence show, without using tables or calculators, that tan 22.5° = √2 - 1.
149. 3(b)(iii)1 mark· Pure Mathematics · Unit 1 Q3 3(b)(iii)show that
150. 3(b)(iii)a)3 marks· Pure Mathematics · Unit 1 Q3 3(b)(iii)a)sin A = sin (B + C)
151. 3(b)(iii)b)2 marks· Pure Mathematics · Unit 1 Q3 3(b)(iii)b)sin A + sin B + sin C = sin (A + B) + sin (B + C) + sin (A + C).
152. 3(c)8 marks· Pure Mathematics · Unit 1 Q3 3(c)Solve the equation cos θ = sin(θ - π/3) for θ.
153. 3(c)6 marks· Pure Mathematics · Unit 1 Q3 3(c)Prove that tan(A + B + C) = (tanA + tanB + tanC - tanA tanB tanC) / (1 - tanA tanB - tanA tanC - tanB tanC).
154. 3(c)6 marks· Pure Mathematics · Unit 1 Q3 3(c)Solve the equation sin θ - √3 cos θ = 1, for -π ≤ θ ≤ π.
155. 3(c)5 marks· Pure Mathematics · Unit 1 Q3 3(c)Without the use of a calculator or tables, find the exact value of cos(π/12).
156. 3(c)6 marks· Pure Mathematics · Unit 1 Q3 3(c)Solve, for 0 ≤ x ≤ π, the equation sin x + sin 3x = 0.
157. 3(c)6 marks· Pure Mathematics · Unit 1 Q3 3(c)Show that, for A ≠ π/2, sec A – tan A = tan (π/4 - A/2).
158. 3(c)(i)3 marks· Pure Mathematics · Unit 1 Q3 3(c)(i)Express f(θ) = 6cosθ + 8sinθ in the form r sin(θ + α) where 0 ≤ α ≤ 90°.
159. 3(c)(i)2 marks· Pure Mathematics · Unit 1 Q3 3(c)(i)Using the formula for sin A + sin B, show that if t = 2 cos θ then sin (n + 1) θ = t sin nθ – sin (n − 1) θ.
160. 3(c)(i)a)3 marks· Pure Mathematics · Unit 1 Q3 3(c)(i)a)Prove that sin((A+B)/2) = cos(C/2).
161. 3(c)(i)b)2 marks· Pure Mathematics · Unit 1 Q3 3(c)(i)b)Prove that sin B + sin C = 2 cos(A/2) cos((B-C)/2).
162. 3(c)(ii)6 marks· Pure Mathematics · Unit 1 Q3 3(c)(ii)Hence, or otherwise, find the general solution of f(θ) = 2.
163. 3(c)(ii)2 marks· Pure Mathematics · Unit 1 Q3 3(c)(ii)Hence, show that sin 3θ = (t² − 1) sin θ.
164. 3(c)(ii)5 marks· Pure Mathematics · Unit 1 Q3 3(c)(ii)Hence, show that sin A + sin B + sin C = 4 cos(A/2) cos(B/2) cos(C/2).
165. 3(c)(iii)7 marks· Pure Mathematics · Unit 1 Q3 3(c)(iii)Using (c) (ii) above, or otherwise, find ALL solutions of sin 3θ = sin θ, 0 ≤ θ ≤ π.
166. 4(a)2 marks· Pure Mathematics · Unit 1 Q4 4(a)Given that θ is an obtuse angle such that sin θ = 2/3, find the value of cos 2θ.
167. 4(a)6 marks· Pure Mathematics · Unit 1 Q4 4(a)By using x = cos²θ, or otherwise, find all values of the angle θ such that 8 cos³θ - 10 cos²θ + 3 = 0, for 0 ≤ θ ≤ π.
168. 4(a)6 marks· Pure Mathematics · Unit 1 Q4 4(a)Solve cos 2θ - 3 cos θ = 1 for 0 ≤ θ < 2π.
169. 4(a)(i)3 marks· Pure Mathematics · Unit 1 Q4 4(a)(i)Given that sinθ = x, show that tanθ = x / √(1 - x²) where 0 < θ < π/2.
170. 4(a)(i)2 marks· Pure Mathematics · Unit 1 Q4 4(a)(i)Express cos 4θ in terms of cos 2θ.
171. 4(a)(i)4 marks· Pure Mathematics · Unit 1 Q4 4(a)(i)Prove that cos 2θ = (1 - tan^2 θ) / (1 + tan^2 θ).
172. 4(a)(i)4 marks· Pure Mathematics · Unit 1 Q4 4(a)(i)Solve the equation cos 3A = 0.5 for 0 ≤ A ≤ π.
173. 4(a)(i)7 marks· Pure Mathematics · Unit 1 Q4 4(a)(i)find the values of R and α correct to one decimal place
174. 4(a)(i)4 marks· Pure Mathematics · Unit 1 Q4 4(a)(i)Show that x = 4 cos θ + 9 sin θ.
175. 4(a)(ii)6 marks· Pure Mathematics · Unit 1 Q4 4(a)(ii)By expressing x in the form r cos (θ – α), where r is positive and 0 < α < π/2, find the MAXIMUM possible value of x.
176. 4(a)(ii)7 marks· Pure Mathematics · Unit 1 Q4 4(a)(ii)Hence, show, without using calculators, that tan 67.5° = 1 + √2.
177. 4(a)(ii)6 marks· Pure Mathematics · Unit 1 Q4 4(a)(ii)Show that cos 3A = 4 cos³ A - 3 cos A.
178. 4(a)(ii)10 marks· Pure Mathematics · Unit 1 Q4 4(a)(ii)Hence, solve the equation cos 4θ + 3 cos 2θ - 1 = 0, for 0 < θ < π.
179. 4(a)(iii)4 marks· Pure Mathematics · Unit 1 Q4 4(a)(iii)The THREE roots of the equation 4p³ – 3p – 0.5 = 0 all lie between -1 and 1. Use the results in (a) (i) and (ii) to find these roots.
180. 4(b)6 marks· Pure Mathematics · Unit 1 Q4 4(b)If cos A = 3/5, find tan (A/2).
181. 4(b)(i)3 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)Obtain an expression for AD in terms of θ.
182. 4(b)(i)2 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)Find, in terms of θ, the length of the side BC.
183. 4(b)(i)3 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)Find, without using tables or calculators, the EXACT values of sin (A + B).
184. 4(b)(i)1 mark· Pure Mathematics · Unit 1 Q4 4(b)(i)Determine the exact value of cos q.
185. 4(b)(i)6 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)Show that tan (α – β) = hx / (x² + d (d+h)).
186. 4(b)(i)5 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)Show that cos 3θ = 4cos³θ - 3cosθ.
187. 4(b)(ii)4 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)Express AD in the form Rcos(θ + α), where R is positive and α is an acute angle.
188. 4(b)(ii)5 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)Find the value of θ if |BC| = 7 cm.
189. 4(b)(ii)3 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)Find, without using tables or calculators, the EXACT values of cos (A - B).
190. 4(b)(ii)1 mark· Pure Mathematics · Unit 1 Q4 4(b)(ii)Determine the exact value of sin p.
191. 4(b)(iii)2 marks· Pure Mathematics · Unit 1 Q4 4(b)(iii)Is 15 a possible value for |BC|? Give a reason for your answer.
192. 4(b)(iii)2 marks· Pure Mathematics · Unit 1 Q4 4(b)(iii)Find, without using tables or calculators, the EXACT values of cos 2A.
193. 4(b)(iii)3 marks· Pure Mathematics · Unit 1 Q4 4(b)(iii)Determine the exact value of sin r.
194. 4(b)(iv)4 marks· Pure Mathematics · Unit 1 Q4 4(b)(iv)Determine the exact value of cos (p + t).
195. 4(c)5 marks· Pure Mathematics · Unit 1 Q4 4(c)Prove that cos⁴A - sin⁴A + 1 = 2 cos²A.
196. 4(c)7 marks· Pure Mathematics · Unit 1 Q4 4(c)Prove that tan (x/2 + π/4) = sec x + tan x.
197. 4(c)(i)5 marks· Pure Mathematics · Unit 1 Q4 4(c)(i)Express cos 3θ in terms of cos θ.
198. 4(c)(i)3 marks· Pure Mathematics · Unit 1 Q4 4(c)(i)Show that (1 - cos 2θ) / sin 2θ = tan θ.
199. 4(c)(ii)6 marks· Pure Mathematics · Unit 1 Q4 4(c)(ii)Hence, solve, for 0 < θ < 2π, the equation cos 3θ + 2 cos 2θ + 4 cos θ + 2 = 0.
200. 4(c)(ii)a)3 marks· Pure Mathematics · Unit 1 Q4 4(c)(ii)a)Show that (1 - cos 4θ) / sin 4θ = tan 2θ.
201. 4(c)(ii)b)2 marks· Pure Mathematics · Unit 1 Q4 4(c)(ii)b)Show that (1 - cos 6θ) / sin 6θ = tan 3θ.
202. 4(d)8 marks· Pure Mathematics · Unit 1 Q4 4(d)Given that sin A = 12/13 and sin B = 4/5, where A and B are acute angles, find cos (A - B) and sin (A + B).
203. 5(a)2 marks· Pure Mathematics · Unit 1 Q5 5(a)Sketch the curve traced out by the cutter.
204. 5(a)(ii)2 marks· Pure Mathematics · Unit 1 Q5 5(a)(ii)Given that sin 2(x + δx) – sin 2x = 2 cos A sin B, find A and B in terms of x and/or δx.
205. 5(b)(i)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)Determine the height of the tide when high tide occurs for the first time.
206. 5(b)(ii)3 marks· Pure Mathematics · Unit 1 Q5 5(b)(ii)Determine the length of time which elapses between the first high tide and the first low tide.