References only, not question text. Open Official paper ↗ where Cambridge publishes the paper,
or see the official Cambridge past-papers page; for other papers, ask your
school. “Paper 42” means Paper 4, variant 2.
Practice questions
Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not
Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.
Q1R-formula · Core · Paper 3 style · 7 marks
(a)Express 3 sin θ + 4 cos θ in the form R sin(θ + α), where R > 0 and 0° < α < 90°. Give the value of α correct to 2 decimal places.[3]
(b)Hence solve the equation 3 sin θ + 4 cos θ = 2 for 0° < θ < 360°.[4]
(a)Express 5 cos x − 12 sin x in the form R cos(x + α), where R > 0 and 0° < α < 90°. Give α correct to 2 decimal places.[3]
(b)State the greatest value of 5 cos x − 12 sin x and find the value of x in the interval 0° < x < 360° at which it occurs.[2]
Show answer
(a) 13 cos(x + 67.38°)
B1 R = 13M1 tan α = 12/5A1 α = 67.38°
(b) 13, at x = 292.62°
B1 13B1 x + 67.38° = 360°, x = 292.62°
Q3R-formula · Standard · Paper 3 style · 7 marks
(a)Express √3 sin θ − cos θ in the form R sin(θ − α), where R > 0 and 0° < α < 90°, giving exact values of R and α.[3]
(b)Hence solve √3 sin 2θ − cos 2θ = 1 for 0° ≤ θ ≤ 180°.[4]
Show answer
(a) 2 sin(θ − 30°)
B1 R = 2M1 tan α = 1/√3A1 α = 30°
(b) θ = 30°, 90°
M1 sin(2θ − 30°) = 1/2A1 2θ − 30° = 30° or 150°M1 divide correctly by 2A1 θ = 30°, 90° and no others
Q4R-formula · Standard · Paper 3 style · 6 marks
(a)Express 7 cos θ + 24 sin θ in the form R cos(θ − α), where R > 0 and 0° < α < 90°. Give α correct to 2 decimal places.[3]
(b)Hence find the greatest and least values of 1/(30 − 7 cos θ − 24 sin θ) as θ varies.[3]
Show answer
(a) 25 cos(θ − 73.74°)
B1 R = 25M1 tan α = 24/7A1 α = 73.74°
(b) Greatest 1/5, least 1/55
M1 denominator ranges from 30 − 25 to 30 + 25A1 greatest 1/5A1 least 1/55
Q5R-formula · Standard · Paper 2 style · 7 marks
(a)Express 2 sin x + 5 cos x in the form R sin(x + α), where R > 0 and 0 < α < π/2. Give R exactly and α in radians correct to 3 decimal places.[3]
(b)Hence solve 2 sin x + 5 cos x = 3 for 0 < x < 2π, giving your answers correct to 3 decimal places.[4]
Show answer
(a) √29 sin(x + 1.190)
B1 R = √29M1 tan α = 5/2A1 α = 1.190
(b) x = 1.360, 5.684
M1 sin(x + 1.190) = 3/√29A1 x + 1.190 = π − 0.5909 = 2.5507 → x = 1.360M1 x + 1.190 = 2π + 0.5909A1 x = 5.684
Q6R-formula · Challenge · Paper 3 style · 9 marks
(a)Show that 6 sin θ cos θ + 8 cos² θ can be written as 3 sin 2θ + 4 cos 2θ + 4.[2]
(b)Express 3 sin 2θ + 4 cos 2θ in the form R sin(2θ + α), where R > 0 and 0° < α < 90°, and hence state the greatest and least values of 6 sin θ cos θ + 8 cos² θ.[3]
(c)Solve 6 sin θ cos θ + 8 cos² θ = 6 for 0° < θ < 180°.[4]
Show answer
(a) Shown
M1 6 sin θ cos θ = 3 sin 2θ and 8 cos² θ = 4(1 + cos 2θ)A1 3 sin 2θ + 4 cos 2θ + 4 (AG)
Q21Trig equations (sec/cosec/cot) · Challenge · Paper 3 style · 6 marks
(a)Prove that cosec 2θ + cot 2θ ≡ cot θ.[3]
(b)Hence solve cosec 4x + cot 4x = 3 for 0° < x < 180°.[3]
Show answer
(a) Proof
M1 (1 + cos 2θ)/sin 2θM1 2 cos² θ/(2 sin θ cos θ)A1 cot θ
(b) x = 9.22°, 99.22°
M1 cot 2x = 3, so tan 2x = 1/3A1 2x = 18.43°, 198.43°A1 x = 9.22°, 99.22°
Q22Trig equations (double angle) · Core · Paper 2 style · 4 marks
(a)Solve the equation sin 2θ = cos θ for 0° ≤ θ ≤ 360°.[4]
Show answer
(a) θ = 30°, 90°, 150°, 270°
M1 2 sin θ cos θ − cos θ = 0M1 cos θ(2 sin θ − 1) = 0A1 90°, 270°A1 30°, 150°
Q23Trig equations (double angle) · Standard · Paper 3 style · 5 marks
(a)Solve the equation 3 cos 2x + 7 cos x + 5 = 0 for 0° < x < 360°.[5]
Show answer
(a) x = 120°, 131.81°, 228.19°, 240°
M1 cos 2x = 2 cos² x − 1A1 6 cos² x + 7 cos x + 2 = 0M1 (2 cos x + 1)(3 cos x + 2) = 0A1 cos x = −1/2: 120°, 240°A1 cos x = −2/3: 131.81°, 228.19°
Q24Trig equations (double angle) · Challenge · Paper 3 style · 5 marks
(a)Solve the equation 2 tan 2θ = 5 tan θ for 0° ≤ θ ≤ 180°.[5]
Show answer
(a) θ = 0°, 24.09°, 155.91°, 180°
M1 4t/(1 − t²) = 5t, t = tan θA1 t = 0 or 4 = 5 − 5t²A1 t² = 1/5A1 θ = 0°, 180°A1 θ = 24.09°, 155.91°
Q25Trig identities · Core · Paper 1 style · 2 marks
(a)Prove the identity (1 − cos² x)/(sin x cos x) ≡ tan x.[2]
Show answer
(a) Proof
M1 1 − cos² x = sin² xA1 sin² x/(sin x cos x) = sin x/cos x = tan x
Q26Trig identities · Standard · Paper 1 style · 6 marks
(a)Prove the identity sin θ/(1 − cos θ) − sin θ/(1 + cos θ) ≡ 2/tan θ.[4]
(b)Hence solve sin θ/(1 − cos θ) − sin θ/(1 + cos θ) = 5 for 0° < θ < 360°.[2]
Show answer
(a) Proof
M1 common denominator 1 − cos² θA1 numerator sin θ(1 + cos θ − 1 + cos θ) = 2 sin θ cos θM1 1 − cos² θ = sin² θA1 2 cos θ/sin θ = 2/tan θ
(b) θ = 21.80°, 201.80°
M1 tan θ = 0.4A1 21.80°, 201.80°
Q27Trig identities · Challenge · Paper 1 style · 7 marks
(a)Prove the identity tan x + 1/tan x ≡ 1/(sin x cos x).[3]
(b)By forming a quadratic equation in tan x, solve tan x + 1/tan x = 4 for 0° < x < 360°, giving exact answers.[4]
Show answer
(a) Proof
M1 sin x/cos x + cos x/sin xM1 (sin² x + cos² x)/(sin x cos x)A1 1/(sin x cos x)
(b) x = 15°, 75°, 195°, 255°
M1 tan² x − 4 tan x + 1 = 0A1 tan x = 2 ± √3A1 x = 75°, 255° (tan x = 2 + √3)A1 x = 15°, 195° (tan x = 2 − √3)
Q28Trig graphs · Core · Paper 1 style · 6 marks
(a)The function f is defined by f(x) = 3 − 2 cos x for 0 ≤ x ≤ 2π. State the range of f.[2]
(b)Describe the shape of the graph of y = f(x), giving the coordinates of its maximum point and its end-points.[2]
(c)Solve the equation f(x) = 4, giving exact answers.[2]
Show answer
(a) 1 ≤ f(x) ≤ 5
B1 least value 1 (at x = 0, 2π)B1 greatest value 5 (at x = π)
(b) Cosine curve reflected and stretched; max (π, 5); ends (0, 1) and (2π, 1)
B1 one 'hump' shape starting and ending at y = 1B1 maximum at (π, 5)
(c) x = 2π/3, 4π/3
M1 cos x = −1/2A1 2π/3, 4π/3
Q29Trig graphs · Standard · Paper 1 style · 7 marks
(a)The graph of y = a + b sin x, for 0 ≤ x ≤ 2π, where a and b are constants and b > 0, has a maximum point at (π/2, 7) and a minimum point at (3π/2, −1). Find a and b.[2]
(b)Find the x-coordinates of the points where the graph crosses the x-axis, giving answers correct to 3 decimal places.[3]
(c)State the set of values of k for which the equation a + b sin x = k has exactly two solutions for 0 ≤ x ≤ 2π.[2]
Show answer
(a) a = 3, b = 4
M1 a + b = 7, a − b = −1A1 a = 3, b = 4
(b) x = 3.990, 5.435
M1 sin x = −3/4A1 π + 0.848 = 3.990A1 2π − 0.848 = 5.435
(c) −1 < k < 7, k ≠ 3
B1 −1 < k < 7B1 k ≠ 3 (three solutions 0, π, 2π when k = 3)
Q30Inverse trig · Standard · Paper 1 style · 6 marks
(a)The function f is defined by f(x) = 2 − 3 sin x for −π/2 ≤ x ≤ π/2. Find the range of f.[2]
(b)Find an expression for f⁻¹(x) and state its domain.[3]
(c)Find the exact value of f⁻¹(0.5).[1]
Show answer
(a) −1 ≤ f(x) ≤ 5
B1 f(π/2) = −1B1 f(−π/2) = 5, range −1 ≤ f(x) ≤ 5
(b) f⁻¹(x) = sin⁻¹((2 − x)/3), −1 ≤ x ≤ 5
M1 x = 2 − 3 sin y, sin y = (2 − x)/3A1 f⁻¹(x) = sin⁻¹((2 − x)/3)B1 domain −1 ≤ x ≤ 5
(c) π/6
B1 sin⁻¹(0.5) = π/6
Similar questions
Questions that test the same subtopic in different years, grouped so you can work on one skill at a time.
References only: find each question in its paper (official link where available, otherwise through your school).
