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TMUA trigonometry questions
All 37 trigonometry questions from the TMUA past papers, 2016 to 2023, newest first, each with a worked solution.
2023
- P1 Q8 · A triangle XYZ is called fun if it has the following properties: angle YXZ = 30^∘ XY = √(3),a YZ =…
- P1 Q9 · How many solutions are there to (1 + 3cos 3θ)² = 4 in the interval 0^∘ ≤ θ ≤ 180^∘?
- P1 Q12 · How many solutions are there to the equation 2^tan^2 x4^sin^2 x = 1 in the range 0 ≤ x ≤ 2π?
- P2 Q12 · In this question, p is a real constant. The equation sin x cos² x = p² sin x has n distinct… hard
2022
- P1 Q1 · How many real solutions are there to the equation 2cos⁴θ - 5cos²θ + 3 = 0 in the interval 0 ≤ θ ≤…
- P1 Q4 · Two sectors of circles are similar. The smaller has radius r and the larger has radius r + 3. The…
- P1 Q9 · This question is about pairs of functions f and g that satisfy f(x) - g(x) = 2sin x f(x),g(x) =… hard
- P1 Q17 · A triangle has a side of length x - 1 opposite an angle of 30^∘, and a side of length -x² + 6x - 5… hard
- P2 Q20 · The functions f₁ to f₅ are defined on the real numbers by f₁(x) = cos x f₂(x) = sin(cos x) f₃(x) =… hard
2021
- P1 Q6 · The function f is given by f(x) = (cos x + 3)/(7 + 5cos x - sin² x) Find the positive difference…
- P1 Q14 · This question uses radians. Find the number of distinct values of x that satisfy the equation… hard
- P1 Q19 · The equation sin²(4^cosθ × 60^∘) = (3)/(4) has exactly three solutions in the range 0^∘ ≤ θ ≤ x^∘… hard
- P2 Q18 · A student chooses two distinct real numbers x and y with 0 The student then attempts to draw a… hard
- P2 Q19 · The angle θ can take any of the values 1^∘, 2^∘, 3^∘, …, 359^∘, 360^∘. For how many of these… hard
2020
- P1 Q12 · How many real solutions are there to the equation 3cos x = √(x) where x is in radians?
- P1 Q18 · Find the number of solutions and the sum of the solutions of the equation 1 - 2cos² x = |cos x|…
- P2 Q2 · Given that tanθ = 2 and 180^∘ < θ < 360^∘, find the value of cosθ.
- P2 Q9 · A student wishes to evaluate the function f(x) = xsin x, where x is in radians, but has a…
- P2 Q15 · Which one of the following is a necessary and sufficient condition for ∑_k=1ⁿsin((kπ)/(3)) =…
2019
- P1 Q14 · x satisfies the simultaneous equations sin 2x + √(3)cos 2x = -1 and √(3)sin 2x - cos 2x = √(3)…
- P1 Q17 · Find the fraction of the interval 0 ≤ θ ≤ π for which the inequality (sin 2θ - 12)(sinθ - cosθ) ≥… hard
- P1 Q19 · Find the value of ∑_k=0⁹⁰ sin(10 + 90k)^∘
2018
- P1 Q6 · Find the number of solutions of the equation xsin 2x = cos 2x with 0 ≤ x ≤ 2π. hard
- P1 Q18 · What is the smallest positive value of a for which the line x = a is a line of symmetry of the…
- P1 Q19 · A triangle ABC is to be drawn with AB = 10 cm, BC = 7 cm and the angle at A equal to θ, where θ is…
- P1 Q20 · Find the value of sin² 0^∘ + sin² 1^∘ + sin² 2^∘ + sin² 3^∘ + ⋯ + sin² 87^∘ + sin² 88^∘ + sin²…
- P2 Q4 · The non-zero real number c is such that the equation cos x = c has two solutions for 0 How many…
- P2 Q14 · In the triangle PQR, PR = 2, QR = p and ∠ RPQ = 30^∘. What is the set of all the values of p for… hard
2017
- P1 Q8 · Find the complete set of values of x, with 0 ≤ x ≤ π, for which (1 - 2sin x)cos x ≥ 0
- P1 Q20 · The lengths of the sides QR, RP and PQ in triangle PQR are a, a + d and a + 2d respectively, where… hard
- P2 Q4 · The following question appeared in an examination: Given that tan x = 3, find the possible values…
- P2 Q12 · The diagram shows the graphs of y = sin 2x and y = cos 2x for -(π)/(2) Which one of the following… hard
2016
- P1 Q8 · Find the maximum angle x in the range 0^∘ ≤ x ≤ 360^∘ which satisfies the equation cos²(2x) +…
- P1 Q10 · How many solutions does the equation xtan x = 1 have in the interval -2π ≤ x ≤ 2π?
- P1 Q17 · It is given that y = (1 + 2cos x)cos 2x for 0 The complete set of values of x for which y is…
- P2 Q3 · What is the value, in radians, of the largest angle x in the range 0 ≤ x ≤ 2π that satisfies the…
- P2 Q17 · Consider these simultaneous equations, where c is a constant: y = 3sin x + 2 y = x + c Which of… hard