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DSE M2 Trigonometric Functions Practice

DSE M2 · 60 practice questions with instant marking and worked solutions.

Trigonometric Functions is an important part of the DSE M2 mathematics curriculum. This topic develops radians, arcs and sectors, compound- and double-angle formulae, product-to-sum formulae and trigonometric equations. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Trigonometric Functions has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 60 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Trigonometric Functions; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 12 of them are DSE Extended Part style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Trigonometric Functions set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Trigonometric Functions is closely linked with the rest of the Extended Part: algebraic and trigonometric skills underpin calculus, and calculus in turn supports probability distributions and statistical inference. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    Convert $240^\circ$ to radians in the form $k\pi$. Find the value of $k$ (as a fraction in lowest terms). Choose the correct answer.
  2. 2 Basic
    A sector has radius 7 cm and angle $\frac{5}{6}\pi$ radians. Write the answer as $k\pi$. Find the value of $k$ for its arc length (as a fraction in lowest terms). Give the answer only.

  3. 3 Basic
    It is given that $\tan A=2$ and $\tan B=\frac{1}{3}$. Find the value of $\tan(A-B)$. Show the essential working and give the answer.

  4. 4 Basic
    $A$ is acute and $\tan A=\frac{15}{8}$. The value of $\tan 2A$ is 30/17. Is this true?
  5. 5 Basic
    Convert $270^\circ$ to radians in the form $k\pi$. Find the value of $k$ (as a fraction in lowest terms). Choose the correct answer.
  6. 6 Basic
    A sector has radius 5 cm and angle $\frac{1}{6}\pi$ radians. Write the answer as $k\pi$. Find the value of $k$ for its area (as a fraction in lowest terms). Give the answer only.

  7. 7 Basic
    It is given that $\tan A=\frac{2}{3}$ and $\tan B=\frac{1}{3}$. Find the value of $\tan(A+B)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  8. 8 Basic
    $A$ is acute and $\tan A=\frac{5}{12}$. The value of $\sin 2A$ is 10/13. Is this true?
  9. 9 Basic
    Convert $210^\circ$ to radians in the form $k\pi$. Find the value of $k$ (as a fraction in lowest terms). Choose the correct answer.
  10. 10 Basic
    A sector has radius 6 cm and angle $\frac{1}{3}\pi$ radians. Write the answer as $k\pi$. Find the value of $k$ for its arc length. Give the answer only.

  11. 11 Basic
    It is given that $\tan A=\frac{2}{3}$ and $\tan B=\frac{1}{2}$. Find the value of $\tan(A+B)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  12. 12 Basic
    $A$ is acute and $\tan A=\frac{7}{24}$. The value of $\cos 2A$ is 14/25. Is this true?
  13. 13 Standard
    Consider the equation $\sin 4x=\frac{1}{2}$ for $0\le x<2\pi$. Find the number of solutions. Choose the correct answer.
  14. 14 Standard
    $2\cos 6x\cos 2x$ is written as $\cos Ax+\cos Bx$ where $A>B>0$. Find the value of $B$. Give the answer only.

  15. 15 Standard
    Convert $72^\circ$ to radians in the form $k\pi$. Find the value of $k$ (as a fraction in lowest terms). Show the essential working and give the answer.

  16. 16 Standard DSE long question
    $A$ is acute and $\tan A=\frac{5}{12}$. Answer each part; give the answer only.
    1. (a) Find $\sin A$ (as a fraction in lowest terms).
    2. (b) Find $\cos A$ (as a fraction in lowest terms).
    3. (c) Find $\sin 2A$ (as a fraction in lowest terms).
    4. (d) Find $\cos 2A$ (as a fraction in lowest terms).
  17. 17 Standard
    Consider the equation $\tan 3x=1$ for $0\le x<\pi$. Find the number of solutions. Choose the correct answer.
  18. 18 Standard
    $2\cos 5x\cos 3x$ is written as $\cos Ax+\cos Bx$ where $A>B>0$. Find the value of $B$. Give the answer only.

  19. 19 Standard
    Convert $30^\circ$ to radians in the form $k\pi$. Find the value of $k$ (as a fraction in lowest terms). Show the essential working and give the answer.

  20. 20 Standard DSE long question
    $A$ is acute and $\tan A=\frac{12}{5}$. Answer each part; give the answer only.
    1. (a) Find $\sin A$ (as a fraction in lowest terms).
    2. (b) Find $\cos A$ (as a fraction in lowest terms).
    3. (c) Find $\sin 2A$ (as a fraction in lowest terms).
    4. (d) Find $\cos 2A$ (as a fraction in lowest terms).
  21. 21 Standard
    Consider the equation $\sin 2x=\frac{1}{2}$ for $0\le x<2\pi$. Find the number of solutions. Choose the correct answer.
  22. 22 Standard
    $2\sin 6x\cos 2x$ is written as $\sin Ax+\sin Bx$ where $A>B>0$. Find the value of $A$. Give the answer only.

  23. 23 Standard
    Convert $300^\circ$ to radians in the form $k\pi$. Find the value of $k$ (as a fraction in lowest terms). Show the essential working and give the answer.

  24. 24 Standard DSE long question
    $\theta$ is acute and $\sin\theta=\frac{4}{5}$. Answer each part; give the answer only.
    1. (a) Find $\cos\theta$ (as a fraction in lowest terms).
    2. (b) Find $\tan\theta$ (as a fraction in lowest terms).
    3. (c) Find $\sin2\theta$ (as a fraction in lowest terms).
    4. (d) Find $\cos2\theta$ (as a fraction in lowest terms).
  25. 25 Advanced
    A sector has radius 7 cm and angle $\frac{2}{3}\pi$ radians. Write the answer as $k\pi$. Find the value of $k$ for its area (as a fraction in lowest terms). Choose the correct answer.
  26. 26 Advanced
    It is given that $\tan A=2$ and $\tan B=\frac{1}{5}$. Find the value of $\tan(A+B)$ (as a fraction in lowest terms). Give the answer only.

  27. 27 Advanced
    $A$ is acute and $\tan A=\frac{4}{3}$. The value of $\sin 2A$ is 24/25. Is this true?
  28. 28 Advanced DSE long question
    $A$ is acute and $\tan A=\frac{15}{8}$. Answer each part; give the answer only.
    1. (a) Find $\sin A$ (as a fraction in lowest terms).
    2. (b) Find $\cos A$ (as a fraction in lowest terms).
    3. (c) Find $\sin 2A$ (as a fraction in lowest terms).
    4. (d) Find $\cos 2A$ (as a fraction in lowest terms).
  29. 29 Advanced
    A sector has radius 11 cm and angle $\frac{2}{3}\pi$ radians. Write the answer as $k\pi$. Find the value of $k$ for its area (as a fraction in lowest terms). Choose the correct answer.
  30. 30 Advanced
    It is given that $\tan A=\frac{1}{3}$ and $\tan B=\frac{3}{4}$. Find the value of $\tan(A+B)$ (as a fraction in lowest terms). Give the answer only.

  31. 31 Advanced
    $A$ is acute and $\tan A=\frac{7}{24}$. The value of $\tan 2A$ is 336/527. Is this true?
  32. 32 Advanced DSE long question
    $A$ is acute and $\tan A=\frac{3}{4}$. Answer each part; give the answer only.
    1. (a) Find $\sin A$ (as a fraction in lowest terms).
    2. (b) Find $\cos A$ (as a fraction in lowest terms).
    3. (c) Find $\sin 2A$ (as a fraction in lowest terms).
    4. (d) Find $\cos 2A$ (as a fraction in lowest terms).
  33. 33 Advanced
    A sector has radius 8 cm and angle $\frac{1}{3}\pi$ radians. Write the answer as $k\pi$. Find the value of $k$ for its arc length (as a fraction in lowest terms). Choose the correct answer.
  34. 34 Advanced
    It is given that $\tan A=\frac{1}{4}$ and $\tan B=\frac{1}{2}$. Find the value of $\tan(A+B)$ (as a fraction in lowest terms). Give the answer only.

  35. 35 Advanced
    $A$ is acute and $\tan A=\frac{3}{4}$. The value of $\cos 2A$ is 7/25. Is this true?
  36. 36 Advanced DSE long question
    $\theta$ is acute and $\sin\theta=\frac{77}{85}$. Answer each part; give the answer only.
    1. (a) Find $\cos\theta$ (as a fraction in lowest terms).
    2. (b) Find $\tan\theta$ (as a fraction in lowest terms).
    3. (c) Find $\sin2\theta$ (as a fraction in lowest terms).
    4. (d) Find $\cos2\theta$ (as a fraction in lowest terms).
  37. 37 Challenging
    Consider the equation $\cos 3x=0$ for $0\le x<2\pi$. Find the number of solutions. Choose the correct answer.
  38. 38 Challenging
    $2\sin 5x\cos 2x$ is written as $\sin Ax+\sin Bx$ where $A>B>0$. Find the value of $A$. Show the essential working and give the answer.

  39. 39 Challenging
    Convert $120^\circ$ to radians in the form $k\pi$. The value of $k$ is 3/2. Is this true?
  40. 40 Challenging DSE long question
    $A$ is acute and $\tan A=\frac{4}{3}$. Answer each part; give the answer only.
    1. (a) Find $\sin A$ (as a fraction in lowest terms).
    2. (b) Find $\cos A$ (as a fraction in lowest terms).
    3. (c) Find $\sin 2A$ (as a fraction in lowest terms).
    4. (d) Find $\cos 2A$ (as a fraction in lowest terms).
  41. 41 Challenging
    Consider the equation $\tan 4x=1$ for $0\le x<\pi$. Find the number of solutions. Choose the correct answer.
  42. 42 Challenging
    $2\sin 8x\sin 5x$ is written as $\cos Bx-\cos Ax$ where $A>B>0$. Find the value of $A$. Show the essential working and give the answer.

  43. 43 Challenging
    A sector has radius 11 cm and angle $\frac{3}{4}\pi$ radians. Write the answer as $k\pi$. The value of $k$ for its area is 33/4. Is this true?
  44. 44 Challenging DSE long question
    $A$ is acute and $\tan A=\frac{7}{24}$. Answer each part; give the answer only.
    1. (a) Find $\sin A$ (as a fraction in lowest terms).
    2. (b) Find $\cos A$ (as a fraction in lowest terms).
    3. (c) Find $\sin 2A$ (as a fraction in lowest terms).
    4. (d) Find $\cos 2A$ (as a fraction in lowest terms).
  45. 45 Challenging
    Consider the equation $\tan 2x=1$ for $0\le x<\pi$. Find the number of solutions. Choose the correct answer.
  46. 46 Challenging
    $2\cos 6x\cos x$ is written as $\cos Ax+\cos Bx$ where $A>B>0$. Find the value of $B$. Show the essential working and give the answer.

  47. 47 Challenging
    Convert $150^\circ$ to radians in the form $k\pi$. The value of $k$ is 6/5. Is this true?
  48. 48 Challenging DSE long question
    $\theta$ is acute and $\sin\theta=\frac{5}{13}$. Answer each part; give the answer only.
    1. (a) Find $\cos\theta$ (as a fraction in lowest terms).
    2. (b) Find $\tan\theta$ (as a fraction in lowest terms).
    3. (c) Find $\sin2\theta$ (as a fraction in lowest terms).
    4. (d) Find $\cos2\theta$ (as a fraction in lowest terms).
  49. 49 Elite
    A sector has radius 7 cm and angle $\frac{3}{4}\pi$ radians. Write the answer as $k\pi$. Find the value of $k$ for its area (as a fraction in lowest terms). Give the answer only.

  50. 50 Elite
    It is given that $\tan A=\frac{1}{3}$ and $\tan B=\frac{1}{5}$. Find the value of $\tan(A+B)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  51. 51 Elite
    $A$ is acute and $\tan A=\frac{20}{21}$. The value of $\tan 2A$ is 840/41. Is this true?
  52. 52 Elite DSE long question
    $A$ is acute and $\tan A=\frac{20}{21}$. Answer each part; give the answer only.
    1. (a) Find $\sin A$ (as a fraction in lowest terms).
    2. (b) Find $\cos A$ (as a fraction in lowest terms).
    3. (c) Find $\sin 2A$ (as a fraction in lowest terms).
    4. (d) Find $\cos 2A$ (as a fraction in lowest terms).
  53. 53 Elite
    A sector has radius 10 cm and angle $\frac{1}{3}\pi$ radians. Write the answer as $k\pi$. Find the value of $k$ for its arc length (as a fraction in lowest terms). Give the answer only.

  54. 54 Elite
    It is given that $\tan A=\frac{2}{3}$ and $\tan B=\frac{1}{3}$. Find the value of $\tan(A-B)$ (as a fraction in lowest terms). Show the essential working and give the answer.

  55. 55 Elite
    $A$ is acute and $\tan A=\frac{8}{15}$. The value of $\sin 2A$ is 240/289. Is this true?
  56. 56 Elite DSE long question
    $A$ is acute and $\tan A=\frac{8}{15}$. Answer each part; give the answer only.
    1. (a) Find $\sin A$ (as a fraction in lowest terms).
    2. (b) Find $\cos A$ (as a fraction in lowest terms).
    3. (c) Find $\sin 2A$ (as a fraction in lowest terms).
    4. (d) Find $\cos 2A$ (as a fraction in lowest terms).
  57. 57 Elite
    A sector has radius 3 cm and angle $\frac{1}{3}\pi$ radians. Write the answer as $k\pi$. Find the value of $k$ for its arc length. Give the answer only.

  58. 58 Elite
    It is given that $\tan A=3$ and $\tan B=\frac{3}{4}$. Find the value of $\tan(A+B)$. Show the essential working and give the answer.

  59. 59 Elite
    $A$ is acute and $\tan A=\frac{4}{3}$. The value of $\cos 2A$ is -7/25. Is this true?
  60. 60 Elite DSE long question
    $\theta$ is acute and $\sin\theta=\frac{3}{5}$. Answer each part; give the answer only.
    1. (a) Find $\cos\theta$ (as a fraction in lowest terms).
    2. (b) Find $\tan\theta$ (as a fraction in lowest terms).
    3. (c) Find $\sin2\theta$ (as a fraction in lowest terms).
    4. (d) Find $\cos2\theta$ (as a fraction in lowest terms).

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Trigonometric Functions

This topic needs particular care with radians, arcs and sectors, compound- and double-angle formulae, product-to-sum formulae and trigonometric equations. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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