TMUA Trigonometry — Practice Questions by Topic

These are SummitPapers original questions, not official past paper questions. Official TMUA questions sorted by topic are on TMUA past papers by topic.

MM4Trigonometry
104 questions67 on Paper 1 · 37 on Paper 2
9 free solutionsThe rest show the correct letter only

Covers: trigonometric equations and the number of solutions, identities and exact values, the sine and cosine rules, radians, arc length and sector area, graphs of sine, cosine and tangent.

Paper 1 (67)Paper 2 (37)Practise this topic

Back to the syllabus · Trigonometry · 1 of 104

Question 1Trigonometry

How many solutions does 2 sin² θ − sin θ − 1 = 0 have in the interval 0 ≤ θ ≤ 2π?

What this topic tests

Trigonometry here means equations and how many solutions they have, identities and exact values, the sine and cosine rules, radians with arc length and sector area, and the graphs of sine, cosine and tangent.

How it is assessed

This is Section 1, on both papers in this set. Each question has five options. A paper is 20 questions in 75 minutes, with no calculator.

This note is written for SummitPapers. The official list of what can be examined is the specification, together with the Notes on Mathematics and, for Paper 2, the Notes on Logic and Proof.

Key methods

How many solutions

On a stated interval, sine and cosine each take a value twice per period, except at the turning values. Tangent has period 180 degrees, or π radians. The interval in the question decides the count.

An identity before a value

sin²θ + cos²θ = 1, and tan θ = sin θ / cos θ where cosine is not zero. An equation that mixes sine and cosine usually becomes one of those before it is solved.

Common mistakes

Degrees in a radian formula

Arc length is rθ and sector area is (1/2)r²θ only when θ is in radians. A degree measure has to be converted first.

Dropping a solution

sin θ = 1/2 is not only 30 degrees. The second solution in the interval is part of the answer, and so is any further period that still lies in the interval.

Worked example

This question is also in the list below, with the solution folded. It is opened here so the method is on the page before the other questions.

How many solutions does 2 sin² θ − sin θ − 1 = 0 have in the interval 0 ≤ θ ≤ 2π?

A. 1
B. 2
C. 3
D. 4
E. 6

Answer: C. 3

Worked solution. Factor the left side as (2 sin θ + 1)(sin θ − 1) = 0, so sin θ = −1/2 or sin θ = 1. In [0, 2π], sin θ = 1 only at θ = π/2, and sin θ = −1/2 at θ = 7π/6 and θ = 11π/6. These three values are distinct. The number of solutions is 3.

Why the other options look right. A counts only θ = π/2. B counts only the two solutions of sin θ = −1/2. D also includes θ = 3π/2, where sin θ = −1, which is not a root. E solves the same equation on 0 ≤ θ ≤ 4π and gets 6 solutions.

Paper 1 questions

Paper 1 is Applications of Mathematical Knowledge. Calculators are not allowed.

Q1. How many solutions does 2 sin² θ − sin θ − 1 = 0 have in the interval 0 ≤ θ ≤ 2π?
Free · worked solution included
Answer and worked solution
Answer: C. 3
Worked solution. Factor the left side as (2 sin θ + 1)(sin θ − 1) = 0, so sin θ = −1/2 or sin θ = 1. In [0, 2π], sin θ = 1 only at θ = π/2, and sin θ = −1/2 at θ = 7π/6 and θ = 11π/6. These three values are distinct. The number of solutions is 3.
Why the other options look right. A counts only θ = π/2. B counts only the two solutions of sin θ = −1/2. D also includes θ = 3π/2, where sin θ = −1, which is not a root. E solves the same equation on 0 ≤ θ ≤ 4π and gets 6 solutions.
Q2. Where it is defined, which expression is equal to (1 − cos 2θ)/sin 2θ?
Free · worked solution included
Answer and worked solution
Answer: A. tan θ
Worked solution. Use 1 − cos 2θ = 2 sin² θ and sin 2θ = 2 sin θ cos θ. The quotient is 2 sin² θ / (2 sin θ cos θ) = sin θ / cos θ, wherever the original expression is defined. That equals tan θ.
Why the other options look right. B inverts the simplified ratio. C cancels 2 sin θ and drops the remaining factor cos θ, leaving sin θ. D writes sin 2θ as sin θ cos θ, without the factor 2, which produces 2 tan θ. E cancels the whole of sin² θ against the single sin θ in the denominator, leaving 1/cos θ = sec θ.
Q3. Solve tan(2x) = 3 tan x for x in the interval 0 ≤ x < π.
Free · worked solution included
Answer and worked solution
Answer: B. x = 0, π/6, 5π/6
Worked solution. Both sides must be defined, so x ≠ π/4, π/2, 3π/4. Let t = tan x. Then tan(2x) = 2t/(1 − t²), so 2t/(1 − t²) = 3t. Multiplying by 1 − t² gives 2t = 3t − 3t³, so 3t³ − t = 0 and t(3t² − 1) = 0. Thus t = 0 or t = 1/√3 or t = −1/√3. In 0 ≤ x < π, tan x = 0 gives x = 0, tan x = 1/√3 gives x = π/6 and tan x = −1/√3 gives x = 5π/6. None of these is an excluded value, so the solutions are x = 0, π/6, 5π/6.
Why the other options look right. A uses tan(2x) = 2t/(1 + t²), with the wrong sign in the denominator; then 2t = 3t + 3t³ gives only t = 0. C divides both sides by tan x and so loses the solution x = 0. D keeps only the positive root t = 1/√3 of t² = 1/3 and misses tan x = −1/√3. E rearranges 3t² = 1 as t² = 3, so it uses tan x = ±√3 and gets π/3 and 2π/3.
Q4. Solve tan x = 2 sin x for x in the interval 0 ≤ x < 2π.
Free · worked solution included
Answer and worked solution
Answer: C. x = 0, π/3, π, 5π/3
Worked solution. The equation needs cos x ≠ 0. Write tan x = sin x/cos x and multiply by cos x: sin x = 2 sin x cos x, so sin x(1 − 2 cos x) = 0. Either sin x = 0, giving x = 0 or π, or cos x = 1/2, giving x = π/3 or 5π/3. None of these has cos x = 0, so all four are valid: x = 0, π/3, π, 5π/3.
Why the other options look right. A divides both sides by sin x and loses the solutions with sin x = 0. B keeps only sin x = 0 and drops the factor 1 − 2 cos x. D takes π/6 as the basic angle for cos x = 1/2, the angle for sin x = 1/2, and then uses x and 2π − x, giving π/6 and 11π/6. E solves cos x = 1/2 with the sine symmetry x and π − x, giving π/3 and 2π/3.
Q5. Solve sin 2x = 1/2 for x in the interval 0° ≤ x < 360°.
Free · worked solution included
Answer and worked solution
Answer: A. 15°, 75°, 195° and 255°
Worked solution. If sin 2x = 1/2 and 0° ≤ x < 360°, then 0° ≤ 2x < 720°. In that range, sin θ = 1/2 at θ = 30°, 150°, 390° and 510°. Dividing by 2 gives x = 15°, 75°, 195° and 255°.
Why the other options look right. B lists only the solutions with 2x still in the first full turn. C solves sin x = 1/2 and does not halve the angle. D replaces 195° by 165°, using 180° − 15°. E replaces sin 2x by 1 − 2 sin² x, the double-angle formula for cos 2x, so 1 − 2 sin² x = 1/2 gives sin x = ±1/2 and x = 30°, 150°, 210° and 330°.
Q6. What is the maximum value of sin x + √3 cos x?
Free · worked solution included
Answer and worked solution
Answer: D. 2
Worked solution. A function a sin x + b cos x has maximum √(a² + b²). Here a = 1 and b = √3, so the maximum is √(1 + 3) = √4 = 2.
Why the other options look right. A takes the maximum of the sin x term alone, 1, and ignores the √3 cos x term. B takes the maximum of the √3 cos x term alone, √3, and ignores the sin x term. C adds those two separate maxima. E forgets the square root in √(a² + b²) and reports a² + b² = 1 + 3 = 4.
Q7. Solve sin 2x = √2 sin x for x in the interval 0° ≤ x < 360°.
Free · worked solution included
Answer and worked solution
Answer: D. 0°, 45°, 180° and 315°
Worked solution. Use sin 2x = 2 sin x cos x. The equation becomes 2 sin x cos x − √2 sin x = 0, so sin x (2 cos x − √2) = 0. Thus sin x = 0 or cos x = √2/2. In the given interval, sin x = 0 at 0° and 180°, and cos x = √2/2 at 45° and 315°. All four values satisfy the original equation, because no division was used. The solutions are 0°, 45°, 180° and 315°.
Why the other options look right. A divides both sides by sin x, which loses the solutions of sin x = 0. B solves only sin x = 0. C solves cos x = √2/2 only in the first quadrant, finding 45° but missing 315°. E factorises with the wrong sign as sin x (2 cos x + √2) = 0, so it uses cos x = −√2/2.
Q8. Which of the following is the general solution of cos(2x) = sin(2x)?
Free · worked solution included
Answer and worked solution
Answer: B. x = π/8 + kπ/2, k any integer
Worked solution. If cos(2x) = 0, then sin(2x) is 1 or −1, so the two sides are not equal. Otherwise divide by cos(2x) to get tan(2x) = 1. Hence 2x = π/4 + kπ for some integer k, and x = π/8 + kπ/2. For k = 0 both sides equal √2/2, and for k = 1 both sides equal −√2/2.
Why the other options look right. A keeps the correct first angle but adds kπ, which is the period of tan before dividing by 2. C divides the period by 2 but leaves the angle as π/4. D is the general solution of tan x = 1, with the double angle ignored. E keeps only the solutions spaced by a full turn, missing the other three families in each interval of length 2π.
Q9. The angles α and β are acute, with tan α = 1/2 and tan β = 1/3. What is the exact value of tan(α − β)?
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Answer
Answer: B. 1/7
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Q10. Which set is the complete solution of sin(2x) = √3 cos x for 0 ≤ x ≤ 2π?
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Answer
Answer: C. π/3, 2π/3, π/2 and 3π/2
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Q11. Find the greatest solution of cos²(3x) + sin(3x) = 5/4 in the range 0° ≤ x ≤ 180°.
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Answer
Answer: E. 170°
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Q12. How many real solutions does the equation sin x = log₁₀ x have?
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Answer
Answer: C. 3
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Q13. Find the complete set of x in the interval 0 < x < π for which (1 − 2 cos x) sin 2x < 0.
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Answer
Answer: B. 0 < x < π/3 or π/2 < x < π
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Q14. Find the complete set of x with 0 ≤ x ≤ π for which (2 sin x − √3)(2 cos x − 1) ≥ 0.
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Answer
Answer: B. x = π/3 or 2π/3 ≤ x ≤ π
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Q15. In triangle ABC the sides AB and AC have lengths 3 and 2, and the side BC has length x, where √7 < x < √13. What is the full range, in degrees, of possible values of angle BAC?
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Answer
Answer: C. 60 < angle BAC < 90
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Q16. How many solutions does x sin 3x = cos 3x have for 0 ≤ x ≤ π?
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Answer
Answer: D. 3
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Q17. Let f(x) = sin(2x/3) + cos(x/2). What is the smallest positive number T such that f(x + T) = f(x) for every real x?
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Answer
Answer: E. 12π
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Q18. A triangle PQR is drawn with PQ = 12 cm, QR = 8 cm and angle P equal to θ. Of the two possible triangles, the larger has twice the area of the smaller. What is cos θ?
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Answer
Answer: D. √10/4
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Q19. Find the value of sin² 5° + sin² 20° + sin² 35° + ... + sin² 155° + sin² 170°.
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Answer
Answer: C. 6
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Q20. The angle x, in degrees, satisfies both sin 2x = √3/2 and cos 2x = 1/2, with 0° ≤ x ≤ 360°. What is the sum of the possible values of x?
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Answer
Answer: E. 240°
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Q21. Find the fraction of the interval 0 ≤ θ ≤ π on which (sin 2θ − 1/2)(cos θ) ≥ 0.
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Answer
Answer: D. 5/6
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Q22. Find the value of the sum from k = 0 to k = 50 of sin(20 + 90k)°.
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Answer
Answer: C. sin 110°
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Q23. Find the number of solutions, and the sum of the solutions, of the equation 2 sin² x = 3|sin x| − 1, where 0° ≤ x ≤ 360°.
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Answer
Answer: B. 6 solutions, sum 1080°
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Q24. Let f(x) = (cos x + 5)/(21 + 9 cos x − sin² x). What is the positive difference between the maximum and minimum values of f(x)?
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Answer
Answer: A. 2/15
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Q25. How many distinct real solutions does cos(2x) = 2x² − 1 have?
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Answer
Answer: D. 2
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Q26. What is the least value of (sin x + 2)(cos x + 2) as x varies over all real numbers?
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Answer
Answer: C. 9/2 − 2√2
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Q27. The equation sin²(45° × 3^{cos θ}) = 1/2 has exactly three solutions in the range 0° ≤ θ ≤ x°. What is the range of all possible values of x?
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Answer
Answer: D. 270 ≤ x < 360
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Q28. How many real solutions does 4 cos⁴ θ − 5 cos² θ + 1 = 0 have for 0 ≤ θ ≤ 2π?
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Answer
Answer: D. 7
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Q29. The solutions of 6x⁴ − 5x² + 1 = 0 are ±cos α and ±cos β. Which equation has solutions ±sin α and ±sin β?
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Answer
Answer: B. 6x⁴ − 7x² + 2 = 0
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Q30. In a triangle, an angle of 30 degrees has opposite side x and another side x² − 6. Find all x for which two non-congruent triangles have these measurements.
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Answer
Answer: C. 3 < x < 1 + √7
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Q31. In triangle PQR, angle P is 30°, PQ = a√2 and QR = a, with a > 0. There are two such triangles, and S has the larger area. Find the ratio of the area of S to the area of the other triangle.
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Answer
Answer: A. (2 + √3) : 1
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Q32. How many solutions does (2 cos 2θ − 1)(2 sin θ − 1) = 0 have for 0° ≤ θ ≤ 360°?
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Answer
Answer: B. 4
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Q33. How many solutions does 16^{sin x} − 6 × 4^{sin x} + 8 = 0 have in the interval 0 ≤ x ≤ 2π?
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Answer
Answer: C. 3
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Q34. How many solutions does √3 sin θ + cos θ = 1 have in the interval 0 ≤ θ ≤ 4π?
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Answer
Answer: D. 5
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Q35. The angle x is measured in radians and satisfies 0 ≤ x ≤ π. What is the total length of the set of x for which sin x ≥ 1/2 and cos 2x ≥ 0?
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Answer
Answer: B. π/6
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Q36. What is the maximum value of 4 sin⁴(2x) + 3 sin²(2x) − 1?
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Answer
Answer: C. 6
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Q37. In a right-angled triangle, the side opposite angle θ has length 5 and the hypotenuse has length 13. What is sin θ?
Free · worked solution included
Answer and worked solution
Answer: B. 5/13
Worked solution. Sine is opposite over hypotenuse, so sin θ = 5/13. The third side is 12, because 5² + 12² = 13², but that side is not needed for sine.
Why the other options look right. A is opposite over adjacent, which is tan θ. C is adjacent over hypotenuse, which is cos θ. D is hypotenuse over opposite, which is 1/sin θ. E is adjacent over opposite, which is 1/tan θ.
Q38. Find the complete set of x, with 0 ≤ x ≤ 2π, for which (2 cos x + √3)(2 sin x − √2) ≤ 0.
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Answer
Answer: C. [0, π/4] ∪ [3π/4, 5π/6] ∪ [7π/6, 2π]
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Q39. Find the sum, in degrees, of all values of x with −180 ≤ x ≤ 180 that satisfy both √3 sin 3x − cos 3x = 0 and sin 3x + √3 cos 3x = 2.
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Answer
Answer: D. 30°
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Q40. What is the maximum value of 11 sin x + 60 cos x?
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Answer
Answer: D. 61
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Q41. What is the maximum value of 48 sin x + 55 cos x?
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Answer
Answer: E. 73
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Q42. What is the greatest value of 20 sin x + 21 cos x?
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Answer
Answer: E. 29
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Q43. Find the greatest value of 12 sin x − 35 cos x for real x.
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Answer
Answer: D. 37
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Q44. Find the greatest value taken by 33 sin x + 56 cos x.
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Answer
Answer: E. 65
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Q45. What is the greatest value taken by 28 sin x + 45 cos x?
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Answer
Answer: C. 53
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Q46. What is the maximum value of 13 sin x + 84 cos x?
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Answer
Answer: A. 85
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Q47. Find the complete set of x with 0 ≤ x ≤ π such that (2 sin x − √3) cos 2x ≥ 0.
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Answer
Answer: B. π/4 ≤ x ≤ π/3 or 2π/3 ≤ x ≤ 3π/4
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Q48. What is the value of sin⁶(15°) + cos⁶(15°)?
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Answer
Answer: E. 13/16
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Q49. How many solutions does the equation sin(2x) = sin x have in the interval 0 ≤ x ≤ 2π?
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Answer
Answer: D. 5
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Q50. Find the minimum value of f(x) = 9 cos⁴(x) − 12 cos²(x) + 7.
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Answer
Answer: B. 3
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Q51. What is the maximum value of 6 sin⁴(x) + 5 sin²(x) − 2?
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Answer
Answer: A. 9
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Q52. Find the complete set of values of x, with −π ≤ x ≤ π, for which (1 − 2 sin x)(1 + 2 cos x) ≤ 0.
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Answer
Answer: C. −π ≤ x ≤ −2π/3, or π/6 ≤ x ≤ 2π/3, or 5π/6 ≤ x ≤ π
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Q53. The angle x, measured in degrees, satisfies both sin 2x + cos 2x = (1 + √3)/2 and sin 2x − cos 2x = (1 − √3)/2, with 0° ≤ x ≤ 360°. What is the sum of the possible values of x?
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Answer
Answer: C. 210°
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Q54. What is the largest solution of 2 sin² x + 5 cos x + 1 = 0 in the range 0 ≤ x < 2π?
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Answer
Answer: C. 4π/3
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Q55. Find the sum of the solutions of tan² x − 4 tan x + 1 = 0 in the range 0 ≤ x < π.
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Answer
Answer: C. π/2
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Q56. How many solutions does 2 sin(cos x) = √3 have in the range 0 ≤ x < 2π?
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Answer
Answer: A. 0
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Q57. Angles in this question are in degrees, and 0 ≤ x ≤ 180. The number x satisfies both 2 sin(3x) − √3 tan(3x) = 0 and 2 sin(3x) + √3 tan(3x) = 2. What is the sum of the possible values of x?
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Answer
Answer: B. 140
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Q58. What fraction of the interval 0 ≤ x ≤ 2π satisfies sin(x + π/6) ≥ √3/2?
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Answer
Answer: A. 1/6
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Q59. Find the greatest value of (1 + sin² x)(3 − 2 sin² x) as x varies over the real numbers.
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Answer
Answer: D. 25/8
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Q60. Find the maximum value of 2(9^{sin x}) − 6(3^{sin x}) + 5.
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Answer
Answer: C. 5
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Q61. Which of these five expressions has the greatest value?
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Answer
Answer: E. 1/cos(5π/12)
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Q62. In triangle ABC, AC = 5 cm, BC = 13 cm and angle ABC is θ. Two such triangles exist, and the larger one has twice the area of the smaller one. What is cos θ?
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Answer
Answer: B. 9√2 / 13
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Q63. In triangle ABC, angle BAC = 60°, BC = 7 cm and AB + AC = 13 cm. What is the area of triangle ABC, in cm²?
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Answer
Answer: C. 10√3
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Q64. The graph of y = sin(cos(2x)) has least positive period P and maximum value M. Which statement is true?
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Answer
Answer: C. P = π and M < 1
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Q65. Which of these five numbers has the least value?
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Answer
Answer: A. √2/2
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Q66. The equation tan(x² + y²) = √3, with angles in radians, describes circles centred at the origin. The disk inside the smallest circle has area P, and the region between each pair of consecutive circles has area Q. What is Q/P?
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Answer
Answer: B. 3
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Q67. For 0 < x < 1/√2, write tan(2 arcsin x) without using trigonometric functions.
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Answer
Answer: D. 2x√(1 − x²)/(1 − 2x²)
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Paper 2 questions

Paper 2 is Mathematical Reasoning. It can test this same topic. Argument, proof, and identifying errors are the topics that appear on Paper 2 only.

Q1. What is the greatest value of (2 sin²(3x) − 5)² as x varies over the reals?
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Answer
Answer: D. 25
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Q2. The number a is positive, and both endpoints of the interval [0, π] are solutions of sin(ax) = 0. The equation has exactly four solutions in that closed interval. What is a?
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Answer
Answer: B. 3
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Q3. The graph of y = sin(2x) is translated π/12 units to the left to give y = g(x). What is the minimum value of g(x) + cos(2x) on the interval 0 ≤ x ≤ π/2?
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Answer
Answer: C. −3/2
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Q4. Find the largest solution of 2 tan² x + 4 = 5/cos x in the interval 0 ≤ x ≤ 2π.
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Answer
Answer: C. 5π/3
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Q5. How many real solutions does the equation 2 sin x − x + 1 = 0 have?
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Answer: C. 1
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Q6. Which one of the following is not true for any real x with −π/2 < x < π/2?
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Answer: D. cos 2x < tan x < sin x for some such x
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Q7. For how many values of x with 0 ≤ x ≤ 2π is 8 sin x + 15 cos x an integer?
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Answer: D. 69
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Q8. In the triangle PQR, PR = 6, QR = p and angle RPQ = 45°. For which values of p are there exactly two possible lengths of PQ?
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Answer: B. 3√2 < p < 6
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Q9. Given that tan θ = 3 and 180° < θ < 360°, what is sin θ?
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Answer: B. −3√10/10
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Q10. A student wants the value of f(x) = 2x cos x, where x is in radians, but the calculator only works in degrees. What could the student type to evaluate f(3) correctly?
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Answer: C. 6 × cos(180 × 3 ÷ π)
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Q11. Which condition is necessary and sufficient for the sum from k = 1 to n of sin(kπ/2) to equal 1?
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Answer: D. n is 1 or 2 more than a multiple of 4
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Q12. The angle θ can be any of 1°, 2°, …, 360°. For how many of these values is sin θ × √(1 − cos² θ) + cos θ × √(1 − sin² θ) equal to 0?
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Answer: B. 2
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Q13. Define f₁(x) = sin x, f₂(x) = sin(sin x), f₃(x) = sin(sin(sin x)) and f₄(x) = sin(sin(sin(sin x))), and let M₁, M₂, M₃ and M₄ be their maximum values. Which statement is true?
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Answer: D. M₁ > M₂ > M₃ > M₄ > 0
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Q14. The equation sin² x = a sin x has n distinct solutions in the interval 0 ≤ x ≤ 2π, where a is a real constant. Which statement is true?
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Answer: B. n = 5 if and only if 0 < |a| < 1
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Q15. How many solutions does x² tan x = sin x have for −π ≤ x ≤ π? Values where tan x is undefined are not counted.
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Answer: E. 5
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Q16. How many solutions does sin⁴ x + cos³ x = 1 have in the interval 0 degrees ≤ x < 360 degrees?
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Answer: D. 3
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Q17. As x varies over the real numbers, what is the least value of (3 sin²x − cos²x − 4 sin x + 6)²?
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Answer: C. 16
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Q18. What fraction of the interval 0 ≤ θ ≤ 2π satisfies (2 + cos θ)(sin θ − √2/2)(1 + sin²θ) ≤ 0?
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Answer: B. 3/4
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Q19. How many solutions does (sin x)²ⁿ = (cos x)ⁿ have in 0° ≤ x ≤ 360°, where n is a positive integer?
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Answer: D. Two when n is odd, and four when n is even
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Q20. Let y = cos 2x + 2 sin x. As x varies over all real numbers, which of the following describes the set of values taken by y?
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Answer: A. −3 ≤ y ≤ 3/2
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Q21. How many solutions does the equation sin(π cos x) = 0 have in the interval 0 ≤ x ≤ 2π?
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Answer: C. 5
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Q22. How many solutions does sin(3x) = sin(x) have for 0 ≤ x ≤ 2π?
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Answer: E. 7
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Q23. How many solutions does the equation cos(3x) = −1/2 have in the interval 0 ≤ x ≤ 2π?
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Answer: C. 6
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Q24. How many solutions does sec²x = 2 tan x + 4 have for 0 ≤ x ≤ 2π?
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Answer: E. 4
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Q25. How many solutions of tan x = sin 2x lie in the closed interval from 0 to 2π?
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Answer: C. 7
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Q26. How many solutions does sin(x + π/3) = sin x have in the interval 0 ≤ x ≤ 2π?
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Answer: D. 2
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Q27. Given that sin θ = −5/13 and 180° < θ < 270°, what is tan θ?
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Answer: C. 5/12
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Q28. The function f is given by f(x) = cos(2x − 100°) + cos(3x − 60°). What is the smallest positive value of a for which the graph of y = f(x) has a line of symmetry at x = a? Give a in degrees.
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Answer: E. 140
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Q29. What is the minimum value of (2 + cos x)/(3 − cos x) for real x?
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Answer: A. 1/4
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Q30. How many solutions does the equation cos(2x) = cos x have in the interval 0 ≤ x ≤ 2π?
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Answer: C. 4
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Q31. A student wants the value of (sin x)/x at x = 3, with x in radians, but the calculator evaluates sine in degrees. Which expression should be entered?
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Answer: D. (1/3) × sin(180 × 3 / π)
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Q32. How many solutions does sin x + sin 2x + sin 3x = 0 have in the interval 0 ≤ x ≤ 2π?
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Answer: D. 7
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Q33. How many solutions does sin²(x) − cos(x) = 1 have for 0° ≤ x ≤ 360°?
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Answer: D. 3
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Q34. As x varies over the real numbers, what is the least value taken by (2 sin² x + 2 sin x + 2)²?
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Answer: C. 9/4
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Q35. For what fraction of the interval 0 ≤ θ ≤ 2π is 4 sin² θ − 1 ≤ 0?
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Answer: C. 1/3
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Q36. Which of the following lists every solution of sin⁴ x − cos⁴ x = cos x in the interval 0° ≤ x ≤ 360°?
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Answer: B. 60°, 180° and 300° only
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Q37. Which of these numbers is the smallest?
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Answer: E. 3/8
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How many solutions does 2 sin2 θ − sin θ − 1 = 0 have in the interval 0 ≤ θ ≤ 2π?

What this topic tests

How it is assessed

Key methods

How many solutions

An identity before a value

Common mistakes

Degrees in a radian formula

Dropping a solution

Worked example

How many solutions does 2 sin2 θ − sin θ − 1 = 0 have in the interval 0 ≤ θ ≤ 2π?

Paper 1 questions

Q1. How many solutions does 2 sin2 θ − sin θ − 1 = 0 have in the interval 0 ≤ θ ≤ 2π?

Q2. Where it is defined, which expression is equal to (1 − cos 2θ)/sin 2θ?

Q3. Solve tan(2x) = 3 tan x for x in the interval 0 ≤ x < π.

Q4. Solve tan x = 2 sin x for x in the interval 0 ≤ x < 2π.

Q5. Solve sin 2x = 1/2 for x in the interval 0° ≤ x < 360°.

Q6. What is the maximum value of sin x + √3 cos x?

Q7. Solve sin 2x = √2 sin x for x in the interval 0° ≤ x < 360°.

Q8. Which of the following is the general solution of cos(2x) = sin(2x)?

Q9. The angles α and β are acute, with tan α = 1/2 and tan β = 1/3. What is the exact value of tan(α − β)?

Q10. Which set is the complete solution of sin(2x) = √3 cos x for 0 ≤ x ≤ 2π?

Q11. Find the greatest solution of cos2(3x) + sin(3x) = 5/4 in the range 0° ≤ x ≤ 180°.

Q12. How many real solutions does the equation sin x = log10 x have?

Q13. Find the complete set of x in the interval 0 < x < π for which (1 − 2 cos x) sin 2x < 0.

Q14. Find the complete set of x with 0 ≤ x ≤ π for which (2 sin x − √3)(2 cos x − 1) ≥ 0.

Q15. In triangle ABC the sides AB and AC have lengths 3 and 2, and the side BC has length x, where √7 < x < √13. What is the full range, in degrees, of possible values of angle BAC?

Q16. How many solutions does x sin 3x = cos 3x have for 0 ≤ x ≤ π?

Q17. Let f(x) = sin(2x/3) + cos(x/2). What is the smallest positive number T such that f(x + T) = f(x) for every real x?

Q18. A triangle PQR is drawn with PQ = 12 cm, QR = 8 cm and angle P equal to θ. Of the two possible triangles, the larger has twice the area of the smaller. What is cos θ?

Q19. Find the value of sin2 5° + sin2 20° + sin2 35° + ... + sin2 155° + sin2 170°.

Q20. The angle x, in degrees, satisfies both sin 2x = √3/2 and cos 2x = 1/2, with 0° ≤ x ≤ 360°. What is the sum of the possible values of x?

Q21. Find the fraction of the interval 0 ≤ θ ≤ π on which (sin 2θ − 1/2)(cos θ) ≥ 0.

Q22. Find the value of the sum from k = 0 to k = 50 of sin(20 + 90k)°.

Q23. Find the number of solutions, and the sum of the solutions, of the equation 2 sin2 x = 3|sin x| − 1, where 0° ≤ x ≤ 360°.

Q24. Let f(x) = (cos x + 5)/(21 + 9 cos x − sin2 x). What is the positive difference between the maximum and minimum values of f(x)?

Q25. How many distinct real solutions does cos(2x) = 2x2 − 1 have?

Q26. What is the least value of (sin x + 2)(cos x + 2) as x varies over all real numbers?

Q27. The equation sin2(45° × 3cos θ) = 1/2 has exactly three solutions in the range 0° ≤ θ ≤ x°. What is the range of all possible values of x?

Q28. How many real solutions does 4 cos⁴ θ − 5 cos² θ + 1 = 0 have for 0 ≤ θ ≤ 2π?

Q29. The solutions of 6x4 − 5x2 + 1 = 0 are ±cos α and ±cos β. Which equation has solutions ±sin α and ±sin β?

Q30. In a triangle, an angle of 30 degrees has opposite side x and another side x2 − 6. Find all x for which two non-congruent triangles have these measurements.

Q31. In triangle PQR, angle P is 30°, PQ = a√2 and QR = a, with a > 0. There are two such triangles, and S has the larger area. Find the ratio of the area of S to the area of the other triangle.

Q32. How many solutions does (2 cos 2θ − 1)(2 sin θ − 1) = 0 have for 0° ≤ θ ≤ 360°?

Q33. How many solutions does 16sin x − 6 × 4sin x + 8 = 0 have in the interval 0 ≤ x ≤ 2π?

Q34. How many solutions does √3 sin θ + cos θ = 1 have in the interval 0 ≤ θ ≤ 4π?

Q35. The angle x is measured in radians and satisfies 0 ≤ x ≤ π. What is the total length of the set of x for which sin x ≥ 1/2 and cos 2x ≥ 0?

Q36. What is the maximum value of 4 sin⁴(2x) + 3 sin²(2x) − 1?

Q37. In a right-angled triangle, the side opposite angle θ has length 5 and the hypotenuse has length 13. What is sin θ?

Q38. Find the complete set of x, with 0 ≤ x ≤ 2π, for which (2 cos x + √3)(2 sin x − √2) ≤ 0.

Q39. Find the sum, in degrees, of all values of x with −180 ≤ x ≤ 180 that satisfy both √3 sin 3x − cos 3x = 0 and sin 3x + √3 cos 3x = 2.

Q40. What is the maximum value of 11 sin x + 60 cos x?

Q41. What is the maximum value of 48 sin x + 55 cos x?

Q42. What is the greatest value of 20 sin x + 21 cos x?

Q43. Find the greatest value of 12 sin x − 35 cos x for real x.

Q44. Find the greatest value taken by 33 sin x + 56 cos x.

Q45. What is the greatest value taken by 28 sin x + 45 cos x?

Q46. What is the maximum value of 13 sin x + 84 cos x?

Q47. Find the complete set of x with 0 ≤ x ≤ π such that (2 sin x − √3) cos 2x ≥ 0.

Q48. What is the value of sin⁶(15°) + cos⁶(15°)?

Q49. How many solutions does the equation sin(2x) = sin x have in the interval 0 ≤ x ≤ 2π?

Q50. Find the minimum value of f(x) = 9 cos4(x) − 12 cos2(x) + 7.

Q51. What is the maximum value of 6 sin4(x) + 5 sin2(x) − 2?

Q52. Find the complete set of values of x, with −π ≤ x ≤ π, for which (1 − 2 sin x)(1 + 2 cos x) ≤ 0.

Q53. The angle x, measured in degrees, satisfies both sin 2x + cos 2x = (1 + √3)/2 and sin 2x − cos 2x = (1 − √3)/2, with 0° ≤ x ≤ 360°. What is the sum of the possible values of x?

Q54. What is the largest solution of 2 sin2 x + 5 cos x + 1 = 0 in the range 0 ≤ x < 2π?

Q55. Find the sum of the solutions of tan2 x − 4 tan x + 1 = 0 in the range 0 ≤ x < π.

Q56. How many solutions does 2 sin(cos x) = √3 have in the range 0 ≤ x < 2π?

Q57. Angles in this question are in degrees, and 0 ≤ x ≤ 180. The number x satisfies both 2 sin(3x) − √3 tan(3x) = 0 and 2 sin(3x) + √3 tan(3x) = 2. What is the sum of the possible values of x?

Q58. What fraction of the interval 0 ≤ x ≤ 2π satisfies sin(x + π/6) ≥ √3/2?

Q59. Find the greatest value of (1 + sin2 x)(3 − 2 sin2 x) as x varies over the real numbers.

Q60. Find the maximum value of 2(9sin x) − 6(3sin x) + 5.

Q61. Which of these five expressions has the greatest value?

Q62. In triangle ABC, AC = 5 cm, BC = 13 cm and angle ABC is θ. Two such triangles exist, and the larger one has twice the area of the smaller one. What is cos θ?

Q63. In triangle ABC, angle BAC = 60°, BC = 7 cm and AB + AC = 13 cm. What is the area of triangle ABC, in cm²?

Q64. The graph of y = sin(cos(2x)) has least positive period P and maximum value M. Which statement is true?

Q65. Which of these five numbers has the least value?

Q66. The equation tan(x² + y²) = √3, with angles in radians, describes circles centred at the origin. The disk inside the smallest circle has area P, and the region between each pair of consecutive circles has area Q. What is Q/P?

Q67. For 0 < x < 1/√2, write tan(2 arcsin x) without using trigonometric functions.

Paper 2 questions

How many solutions does 2 sin² θ − sin θ − 1 = 0 have in the interval 0 ≤ θ ≤ 2π?

How many solutions does 2 sin² θ − sin θ − 1 = 0 have in the interval 0 ≤ θ ≤ 2π?

Q1. How many solutions does 2 sin² θ − sin θ − 1 = 0 have in the interval 0 ≤ θ ≤ 2π?

Q11. Find the greatest solution of cos²(3x) + sin(3x) = 5/4 in the range 0° ≤ x ≤ 180°.

Q12. How many real solutions does the equation sin x = log₁₀ x have?

Q19. Find the value of sin² 5° + sin² 20° + sin² 35° + ... + sin² 155° + sin² 170°.

Q23. Find the number of solutions, and the sum of the solutions, of the equation 2 sin² x = 3|sin x| − 1, where 0° ≤ x ≤ 360°.

Q24. Let f(x) = (cos x + 5)/(21 + 9 cos x − sin² x). What is the positive difference between the maximum and minimum values of f(x)?

Q25. How many distinct real solutions does cos(2x) = 2x² − 1 have?

Q27. The equation sin²(45° × 3^{cos θ}) = 1/2 has exactly three solutions in the range 0° ≤ θ ≤ x°. What is the range of all possible values of x?

Q29. The solutions of 6x⁴ − 5x² + 1 = 0 are ±cos α and ±cos β. Which equation has solutions ±sin α and ±sin β?

Q30. In a triangle, an angle of 30 degrees has opposite side x and another side x² − 6. Find all x for which two non-congruent triangles have these measurements.

Q33. How many solutions does 16^{sin x} − 6 × 4^{sin x} + 8 = 0 have in the interval 0 ≤ x ≤ 2π?

Q50. Find the minimum value of f(x) = 9 cos⁴(x) − 12 cos²(x) + 7.

Q51. What is the maximum value of 6 sin⁴(x) + 5 sin²(x) − 2?

Q54. What is the largest solution of 2 sin² x + 5 cos x + 1 = 0 in the range 0 ≤ x < 2π?

Q55. Find the sum of the solutions of tan² x − 4 tan x + 1 = 0 in the range 0 ≤ x < π.

Q59. Find the greatest value of (1 + sin² x)(3 − 2 sin² x) as x varies over the real numbers.

Q60. Find the maximum value of 2(9^{sin x}) − 6(3^{sin x}) + 5.

Q1. What is the greatest value of (2 sin²(3x) − 5)² as x varies over the reals?

Q13. Define f₁(x) = sin x, f₂(x) = sin(sin x), f₃(x) = sin(sin(sin x)) and f₄(x) = sin(sin(sin(sin x))), and let M₁, M₂, M₃ and M₄ be their maximum values. Which statement is true?

Q15. How many solutions does x² tan x = sin x have for −π ≤ x ≤ π? Values where tan x is undefined are not counted.

Q16. How many solutions does sin⁴ x + cos³ x = 1 have in the interval 0 degrees ≤ x < 360 degrees?

Q19. How many solutions does (sin x)²ⁿ = (cos x)ⁿ have in 0° ≤ x ≤ 360°, where n is a positive integer?

Q33. How many solutions does sin²(x) − cos(x) = 1 have for 0° ≤ x ≤ 360°?

Q34. As x varies over the real numbers, what is the least value taken by (2 sin² x + 2 sin x + 2)²?

Q35. For what fraction of the interval 0 ≤ θ ≤ 2π is 4 sin² θ − 1 ≤ 0?

Q36. Which of the following lists every solution of sin⁴ x − cos⁴ x = cos x in the interval 0° ≤ x ≤ 360°?