Math topic中文

Sine cosine rule

use of sine rule and cosine rule for any triangle

Practice questions

  1. 1The diagram shows the positions of a harbour, H, a lighthouse, L, and two buoys P and Q. HPQ is a straight line. The bea…
  2. 2A circle with centre O has a tangent XY at D. ∠ODC = 70°, ∠BCD = 120° and ∠ODA = 15°. Find ∠CDY. Find ∠DAB. Find ∠OBA. T…
  3. 3In 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…
  4. 4In triangle ABC, AB = 14.6 cm, BC = 7.6 cm and angle BAC = 31°. Find obtuse angle ACB. Answer ...° area of triangle ABC.…
  5. 5The 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…
  6. 6在△ABC\triangle ABC中，若b2=acb^{2}=ac，B=60∘B=60^{\circ}，则A=A=( ) A. 120∘120^{\circ} B. 60∘60^{\circ} C. 45∘45^{\circ} D.…
  7. 7The 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…
  8. 8The diagram shows a parallelogram ABCD on horizontal ground where AC is a path. AB = 15 m and BC = 8 m. The bearing of B…
  9. 9The area of triangle ABC is 7.5 cm^2. Two of the sides are of length 5 cm and 6 cm respectively. Find the length of the…
  10. 10The diagram above shows a park ABCDABCD that is on horizontal ground, crossed by a straight path ACAC. Point BB is due no…
  11. 11The 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…
  12. 12Airport B is 500 km away from airport A on a bearing of 064°. An aircraft leaves airport A at 22 45 on Monday to fly to…
  13. 13The diagram shows a goat tied to a pole at A. The length of the rope attached to the goat is 6 m. A is due west of the c…
  14. 14The diagram shows the positions of four checkpoints W, X, Y and Z in a jungle reserve area. The checkpoints are connecte…
  15. 15In the parallelogram PQRS, QR = 12.7 cm and angle PQR = 50°. Show that the length of RS = 5.407 cm. Answer Hence or othe…
  16. 16An aircraft waiting to land is flying around a triangular circuit ABC. A, B and C are vertically above three beacons, X,…
  17. 17P, 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…
  18. 18The 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…
  19. 19设△ABC\triangle ABC的内角AA，BB，CC所对边的长分别为aa，bb，c.c.若sin⁡A=2sin⁡B\sin A=2\sin B，c=4,C=π3c=4,C= \dfrac {π}{3}，则△ABC\triangle ABC…
  20. 20O is the centre of the equilateral triangle. The triangle ABC forms the base of a pyramid. The vertex, X is vertically a…
  21. 21In the diagram, OAB is a sector of a circle with centre O. Given that OA = 12 cm, ∠AOB = θ\theta radians, OC = 8 cm and…
  22. 22已知a→=(2λsin⁡x,sin⁡x+cos⁡x) \overrightarrow{a}=(2λ\sin x,\sin x+\cos x)，b→=(3cos⁡x,λ(sin⁡x−cos⁡x))(λ>0) \overrightarrow{b}=( \sqrt {3}\cos x,λ(\sin x-\cos x))(λ > 0)，函数…
  23. 23The 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…
  24. 24ABL 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…
  25. 25The 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}…
  26. 26A, 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…
  27. 27在锐角△ABC\triangle ABC中，角AA，BB，CC所对的边分别为aa，bb，cc，已知a=bcos⁡C+33csin⁡Ba=b\cos C+ \dfrac { \sqrt {3}}{3}c\sin B． (1)若a=2a=2…
  28. 28The diagram shows the positions of three towns, A, B and C. AB is 73 km, BC is 62 km and AC is 56 km. A is due North of…
  29. 29在△ABC\triangle ABC中，角AA，BB，CC的对边分别为aa，bb，cc，若a2=b2+14c2a^{2}=b^{2}+ \dfrac {1}{4}c^{2}，则acos⁡Bc= \dfrac {a\cos B}{c}=…
  30. 30The diagram represents a flat plot of land PQRS. Find the area of the land PQRS. Answer ___ m^2 the length of PS. Answer…
  31. 31A container is a prism with a cross section in the shape of a right-angled triangle. The container has a height of 12 cm…
  32. 32设锐角△ABC\triangle ABC的三个内角AA，BB，CC的对边分别为aa，bb，cc，且c=1c=1，A=2CA=2C，则△ABC\triangle ABC周长的取值范围为()(\quad)…
  33. 33The diagram shows a field ABCD on horizontal ground. Angle ABC = 115°, Angle CAD = 58°, AB = 9 m, BC = 6 m and CD = 11 m…
  34. 34在△ABC\triangle ABC中，A=60∘A=60^{\circ}，AB=2AB=2，且△ABC\triangle ABC的面积S△ABC=32S_{\triangle ABC}= \dfrac { \sqrt {3}}{2}，则边BCBC…
  35. 35Points 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°…
  36. 36In the diagram below, triangle ABC is congruent to triangle PNA. Given that BC = 22 cm, AP = 32 cm and angle ACB = 91°,…
  37. 37P, Q, R and S are points on a triangular garden. S R Q is a straight line. It is given that PQ=55 mPQ = 55\ \text{m}, RQ=40 mRQ = 40\ \text{m}…
  38. 38In Diagram I below, ABCDE is a regular pentagon, centre O. OA=OB=4 cmOA = OB = 4\text{ cm}. (Diagram I and Diagram II are shown…
  39. 39The diagram shows three points, A, B and L on a horizontal field. AL=37.5 m,AB=60 mAL = 37.5\ \mathrm{m}, AB = 60\ \mathrm{m} and ∠LAB=63∘\angle LAB = 63^\circ…
  40. 40在△ABC\triangle ABC中，已知3sin⁡2B=1−cos⁡2B \sqrt {3}\sin 2B=1-\cos 2B． (1)求角BB的值； (2)若BC=2BC=2，A=π4A= \dfrac {π}{4}，求△ABC\triangle ABC…
  41. 41ABCD is a field on horizontal ground. AD = 168 m, BD = 325 m and CD = 151 m. The bearing of B from A is 155° and the bea…
  42. 42(1)三角形ABCABC中，AB=23,BC=2,∠C=600AB=2\sqrt{3},BC=2,\angle C={{60}^{0}}，则三角形ABCABC的面积为___ (2)设等比数列{an}\left\{ {{a}_{n}} \right\}满足 a  ⁣1+a  ⁣3=10a{\,\!}_{1}+a{\,\!}_{3}=10…
  43. 43解三角形
  44. 44设a→=(2cos⁡x+23sin⁡x,1) \overrightarrow{a}=(2\cos x+2 \sqrt {3}\sin x,1)，b→=(cos⁡x,−y) \overrightarrow{b}=(\cos x,-y)满足a→⋅b→=0 \overrightarrow{a}⋅ \overrightarrow{b}=0…
  45. 45设双曲线CC：x2a2−y2b2=1(a>0,b>0)\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\left(a>0,b>0\right)的左、右焦点分别为F1{{F}_{1}}，F2{{F}_{2}}…
  46. 46The diagram shows part of a map of a small town. Joel's house is located at point W, the childcare centre at point X, th…
  47. 47已知钝角三角形ABCABC的面积是12 \dfrac {1}{2}，c=1c=1，a=2a= \sqrt {2}，则b=b= ___ ．
  48. 48In the diagram, not drawn to scale, A, B, C, D and E are points on the circle. CPE and BPD are straight lines. It is giv…
  49. 49ABCD is a field on horizontal ground. AD = 163 m, BD = 329 m, CD = 152 m and angle BDC = 48°. The bearing of B from A is…
  50. 50Three points A, B and C lie on a horizontal ground are such that AB = 24 m and BC = 53 m. Point B is due north of C. The…
  51. 51W, X, Y and Z are points on a horizontal ground and PW is a vertical flag pole. WX=7 mWX = 7\ \text{m}, WZ=3 mWZ = 3\ \text{m},…
  52. 52在△ABC\triangle ABC 中，已知 a=5a=5， b=7b=7， c=8c=8，则 A+C=A+C=( ) A. 150∘150^{\circ} B. 135∘135^{\circ} C. 120∘120^{\circ}…
  53. 53设F1F_{1}，F2F_{2}是双曲线CC：x2a2−y2b2=1(a>0,b>0)\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1(a>0,b>0)的左右焦点，PP是双曲线CC…