Math topic中文

Trig ratios

use of trigonometric ratios (sine, cosine and tangent) of acute angles to calculate unknown sides and angles in right-angled triangles

Practice questions

  1. 1In the diagram, there are two right-angled triangles ΔABD and ΔCEB such that AD = 12 cm, BD = 15 cm and BE = 20 cm. Find…
  2. 2In triangle ABC, ∠ABC=90∘\angle ABC = 90^\circ, BX=2XCBX = 2XC and sin⁡∠AXC=35\sin \angle AXC = \dfrac{3}{5}. Express as a fraction in its…
  3. 3The diagram shows the positions of a harbour, H, a lighthouse, L, and two buoys P and Q. HPQ is a straight line. The bea…
  4. 4A piece of wood which was 50 cm long, was in the shape of a uniform cylinder. A prism is cut out from the wood such that…
  5. 5The points A, B, C, D and E are shown in the diagram below such that ABC and AED are straight lines and BE = 8\sqrt{8}…
  6. 6In the diagram, BA and BC are tangents to the circle with centre O. BO meets the circle at D, OA = 10 cm and BC = 28 cm.…
  7. 7In the diagram, PQRS is a vertical rectangular wall of height 2.5 m and length 7.5 m. T is a point on level ground from…
  8. 8The diagram shows four points P, Q, R and S which lie on level ground in a garden. P is due south of Q. The bearing of S…
  9. 9在平面直角坐标系中，点OO为坐标原点，直线y=kx+k(k>0)y=kx+k(k > 0)与xx轴交于点AA，与yy轴交于点BB，过点AA的抛物线y=x2+bx+cy=x^{2}+bx+c与xx轴交于另一点P(1)P(1)…
  10. 10In the diagram, ABCDEFGH is a cuboid with dimensions 8 cm by 6 cm by 15 cm. V is the centre of the rectangular base. Sho…
  11. 11Solve the equation 2cosec⁡(x−π3)=−42\cosec\left(x-\dfrac{\pi}{3}\right)=-4 for 0≤x≤2π0\le x\le 2\pi. Solve the equation cot⁡(2x+10∘)=tan⁡70∘\cot(2x+10^\circ)=\tan70^\circ…
  12. 12sin⁡x∘=0.66913\sin x^{\circ} = 0.66913 Given that xx is an obtuse angle, find xx. Answer x = ... cos⁡y∘=−cos⁡121∘\cos y^{\circ} = -\cos 121^{\circ}…
  13. 13The figure above shows a solid. The cross-section of the solid is a sector of a circle of radius 5 cm and angle 30°. The…
  14. 14在△ABC\triangle ABC中，∠C=90°∠C=90°，tan⁡A=43\tan A=\dfrac{4}{3}，AB=20AB=20，则AC=AC=___.
  15. 15Triangle PQR is a right-angled triangle with PQ = 10 cm, QR = 6 cm and ∠PRQ = 90°. M is a point on PR such that MR = 5 P…
  16. 16The diagram shows a parallelogram ABCD on horizontal ground where AC is a path. AB = 15 m and BC = 8 m. The bearing of B…
  17. 17The diagram shows a semi-circle BAC, with diameter BC = 34 cm. AD and BCD are straight lines. Given that cos⁡∠ABC=817\cos\angle ABC = \dfrac{8}{17}…
  18. 18The diagram shows a circle C1C_1 with centre P(5,3)P(5, 3). The circle touches the x-axis, the tangent OQ and another line Q…
  19. 19The diagram above shows a park ABCDABCD that is on horizontal ground, crossed by a straight path ACAC. Point BB is due no…
  20. 20在Rt△ABC中，各边的长度都扩大两倍，那么锐角A的各三角函数值（ ） A. 都扩大两倍 B. 都缩小两倍 C. 不变 D. 都扩大四倍
  21. 21The diagram shows the positions of 3 points A, B, and C, at sea level. AB = 4.83 km and BC = 7.24 km. The bearing of B f…
  22. 22A, B and C are points (6, 10), (6,6) and (q, 2) respectively. The area of triangle ABC is 16 square units. Show that q =…
  23. 23Given that 0<x<1800 < x < 180, find the possible values of xx for sin⁡x∘=0.985\sin x^{\circ} = 0.985. Give your answers correct to 1 d…
  24. 24In the diagram, the points P, Q and R have coordinates (9, -3), (6,1) and (18, -3) respectively. Find the length of PQ.…
  25. 25已知F1F_{1}，F2F_{2}分别是双曲线C:x2a2−y2b2=1(a>0,b>0)C:\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1(a>0,b>0)的左、右焦点，PP是该双曲线右支…
  26. 26In the diagram, BDCE is a straight line. BD = 4 cm, AC = 10 cm and AB = AD. Given that the area of triangle ABD is 16 cm…
  27. 27In the diagram, a wooden ramp is shown where ∠BCF=90∘\angle BCF = 90^\circ and ∠CBF=35∘\angle CBF = 35^\circ. ABCD, CDEF and ABFE ar…
  28. 28In the parallelogram PQRS, QR = 12.7 cm and angle PQR = 50°. Show that the length of RS = 5.407 cm. Answer Hence or othe…
  29. 29The diagram shows a rectangular table ABCD arranged such that the side AD is inclined at an angle of θ\theta to the wal…
  30. 30The diagram shows a right-angled triangle AOB where AO = 15 cm and BO = 8 cm. P lies on AB such that OP is an arc of a c…
  31. 31在△ABC\triangle ABC中，∠C=90°∠C=90°，若cos⁡A=13\cos A=\dfrac{1}{3}，则sin⁡B\sin B的值为___；
  32. 32The diagram shows a semicircle, centre O, radius 16 cm. Show that angle AOB = 1.287 radians, correct to 3 decimal places…
  33. 33P, Q, R and S are four villages located on an island, where Q is due north of R. PQ = 924 m, QR = 640 m, PR = 1124 m, RS…
  34. 34The figure shows two rods PA and PB which are hinged at P. The rods slide such that P, A and B are on the same vertical…
  35. 35The sine of an obtuse angle is 5/135/13. Without the use of a calculator, find the value for the cosine of the same angle.…
  36. 36The diagram shows four points P, Q, R and S on level ground. Q is due east of P. Angle SPQ = 40° and angle PSR = 15°. PQ…
  37. 37O is the centre of the equilateral triangle. The triangle ABC forms the base of a pyramid. The vertex, X is vertically a…
  38. 38A logo, as shown below, has a circle with centre O and radius 2 cm, an equilateral triangle ABC of side 5 cm and an isos…
  39. 39The diagram below shows an indoor adventure park in the shape of a trapezoidal prism with a square base ABCD. The indoor…
  40. 4022 P, Q and R are three points on level ground such that RPQ forms a right-angled triangle with PQ and RQ measuring 135…
  41. 41The diagram shows a lecture theatre in the shape of a prism with height 9 metres. The floor of the lecture theatre, ABCD…
  42. 42The diagram shows the cross section of a plastic pipe. The arc AB is part of the circle with centre O and radius 45 mm.…
  43. 43The diagram below shows three points, A, B and C on flat ground. AB = 80 m, AC = 100 m and BC = 90 m. C is due east of A…
  44. 44In the diagram, OAPB is a sector of circle with centre O and radius 9 cm. Angle AOB = 1.4 radians. C is the centre of th…
  45. 45观察下列等式 ①sin⁡30∘=12cos⁡60∘=12②sin⁡45∘=22cos⁡45∘=22③sin⁡60∘=32cos⁡30∘=32…①\sin 30^{\circ}= \dfrac {1}{2}\cos 60^{\circ}= \dfrac {1}{2}②\sin 45^{\circ}= \dfrac { \sqrt {2}}{2}\cos 45^{\circ}= \dfrac { \sqrt {2}}{2}③\sin 60^{\circ}= \dfrac { \sqrt {3}}{2}\cos 30^{\circ}= \dfrac { \sqrt {3}}{2}……
  46. 46应用计算器填一填，分别比较各个三角函数值的大小，说一说有什么规律： （1）cos20°= ，cos40°= ，cos60°= ；cos80°= ；规律： （2）tan10°= ，tan30°= ，tan50°= ；tan70°= ． 规律：…
  47. 47ABL is a park on horizontal ground. A is 1390 m from L on a bearing of 076°. B is 960 m from L on a bearing of 138°. Fin…
  48. 48The diagram shows a triangle ABC with AB produced to D. AB=18 cmAB = 18\,\mathrm{cm}, BC=20 cmBC = 20\,\mathrm{cm}, AC=22 cmAC = 22\,\mathrm{cm}…
  49. 49A, B, C and D are points on horizontal ground. AB = 35 m and BC = 68 m. The bearing of B from A is 137∘137^\circ and the b…
  50. 50The diagram represents a flat plot of land PQRS. Find the area of the land PQRS. Answer ___ m^2 the length of PS. Answer…
  51. 51The triangle with vertices A (2, 5), B (−2, −3) and C (2, −2) is shown in the diagram below. Find (a) the area of triang…
  52. 52A container is a prism with a cross section in the shape of a right-angled triangle. The container has a height of 12 cm…
  53. 53A chock is a block placed against a wheel or rounded object, to prevent it from moving. Jenny had to come up with a prot…
  54. 54The diagram shows a regular 12-sided polygon inscribed in a circle and a right-angled triangle PQR with length QR=6QR = 6…
  55. 55已知∠A+∠B=90°，sinB=0.8436，那么cosA=
  56. 56Given that cos⁡(A+B)=−513\cos(A+B)= -\dfrac{5}{13} and sin⁡Asin⁡B=1526\sin A\sin B=\dfrac{15}{26}, where A and B are acute angles and angle A > a…
  57. 57Figure 1 shows an open container in the form of an isosceles trapezoidal prism ABCDEFGH with square base ABCD of sides xx…
  58. 58The diagram shows five towns A, B, C, D and E. Towns B, C, D lie on a straight line. Town E is due east of Town D. It is…
  59. 59Points A, B, C and D are at sea level. AD = 980 km, AB = 710 km and CD = 1100 km. The bearing of B and C from A are 148°…
  60. 60The diagram shows a circle, centre O, radius 10 cm. AB is a chord of length 16 cm. Show that angle AOB = 1.855 radians,…