Math topic中文

Six functions

six trigonometric functions for angles of any magnitude (in degrees or radians)

Practice questions

  1. 1Given that cos⁡x=−45\cos x = -\frac{4}{5} and tan⁡x>0\tan x > 0, find sin⁡2x\sin 2x, cos⁡x2\cos \frac{x}{2}.
  2. 2A bird is sitting on the top, TT, of a vertical pole 20 m above the ground and the angle of elevation of the bird at TT…
  3. 3Given that sin⁡A=−35\sin A = -\frac{3}{5} and cos⁡B=−817\cos B = -\frac{8}{17}, where angle AA and angle BB are in the same quadrant,…
  4. 4The diagram shows a chocolate bar in the form of a triangular prism and the cross-section of the chocolate bar is an iso…
  5. 5The diagram shows two chords AB and CD intersect at X such that angle CXB = 90∘90^\circ. Given that AC=5 cmAC = 5\ \text{cm},…
  6. 6The diagram shows two triangles DAB and DBC. It is given that BD = 5 m and BC = 2 m. Angle DAB = angle DBC = angle DEC =…
  7. 7A particle, P, starts moving in a straight line with a velocity of 8 m/s at a displacement of 3 m from a fixed point O.…
  8. 8A 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…
  9. 9The diagram shows a trapezium ABCDABCD. The point XX lies on line ABAB such that DX=12 cmDX = 12\text{ cm} and CX=9 cmCX = 9\text{ cm}…
  10. 10The diagram below shows two horizontal lines ABAB and CDCD, where LL is the vertical distance between the lines ABAB an…
  11. 11The diagram below shows an isosceles triangle ABCABC, where AB=AC=2 cmAB = AC = 2\text{ cm}, and a square BCDEBCDE. MM is the midp…
  12. 12Prove the trigonometric identity: csc⁡2A+2cot⁡A(cos⁡A+sin⁡A)2=csc⁡2A\dfrac{\csc^{2}A + 2\cot A}{(\cos A + \sin A)^{2}} = \csc^{2}A Hence solve the equ…
  13. 13Without using the calculator, simplify the following trigonometric functions. sin⁡(x+4π3)\sin \left(x + \frac{4\pi}{3}\right) cos⁡75∘\cos 75^\circ…
  14. 14The diagram shows part of the curve y=4x+5y = \sqrt{4x + 5}. The normal of the curve at PP meets the xx-axis at TT. The xx…
  15. 15判断以下说法是否正确((均指在平面直角坐标系中，始边在xx轴正半轴上).(1)).(1)第一象限角一定是锐角;(2);(2)终边相同的角一定相等;(3);(3)小于90∘90^{\circ}的角一定是锐角;(4);(4)钝角的终…
  16. 16In the diagram, BE intersects AD at F. AF = 7 cm and FD = 4 cm. ∠ABF\angle ABF, ∠ACD\angle ACD and ∠DEF\angle DEF are right an…
  17. 17The diagram shows a pair of similar triangles YUXYUX and VUWVUW. Point YY lies on the straight line VUVU such that VY=12 cmVY = 12\text{ cm}…
  18. 1812 Solve 0.5sin⁡2x=3sin⁡x0.5\sin 2x = 3\sin x for 0<x<2π0 < x < 2\pi. Solve, for values of θ\theta between 0∘0^\circ and 180∘180^\circ, csc⁡(θ2−40∘)+5=0.\csc\left(\frac{\theta}{2}-40^\circ\right)+5=0.…
  19. 19Show that ddx(cot⁡x)=−csc⁡2x\dfrac{d}{dx}(\cot x) = -\csc^{2} x . Hence find the value of ∫π/4π/2−cot⁡2x dx\displaystyle \int_{\pi/4}^{\pi/2} -\cot^{2} x\, dx…
  20. 20The diagram shows the side view of a 14 cm by 8 cm rectangular block PQRSPQRS, placed on a ramp, VSVS, tilted at an acute…
  21. 21Given that sin⁡A=−45\sin A = -\frac{4}{5}, tan⁡B=−512\tan B = -\frac{5}{12} and cos⁡A>0\cos A > 0, where AA and BB are in different quadra…
  22. 22In triangle ABCABC shown below, sides ABAB and ACAC are (p−62) cm(p - 6\sqrt{2})\text{ cm} and (4−2) cm(4 - \sqrt{2})\text{ cm} respec…
  23. 23The figure shows a piece of cardboard in the shape of a trapezium in which ∠BAD\angle BAD and ∠ABC\angle ABC are right angle…
  24. 24In the diagram below, the line OPOP makes an angle of α∘\alpha^\circ with the positive xx-axis such that tan⁡α=0.2\tan \alpha = 0.2…
  25. 25State the range of tan⁡−1p\tan^{-1} p. Given that cos⁡−1k=x\cos^{-1} k = x, where 0≤x≤π20 \le x \le \frac{\pi}{2}, find the principal va…
  26. 26Prove the identity 1+cos⁡θsin⁡θ+sin⁡θ1+cos⁡θ=2csc⁡θ\frac{1+\cos \theta}{\sin \theta} + \frac{\sin \theta}{1+\cos \theta} = 2\csc \theta. Hence, find a…
  27. 27State the principal value of sin⁡−1(−32),thevaluesbetweenwhichtheprincipalvalueof\sin^{-1}\left(-\dfrac{\sqrt{3}}{2}\right), the values between which the principal value of …
  28. 28A and B are acute angles, such that cos⁡A=35\cos A = \frac{3}{5} and tan⁡B=125\tan B = \frac{12}{5}.\n\nFind the exact values of tan⁡2A\tan 2A…
  29. 29By writing 105∘105^\circ as a sum of two angles, without using a calculator, show that sin⁡105∘=24(3+1).\sin 105^\circ = \frac{\sqrt{2}}{4}(\sqrt{3} + 1).…
  30. 30Show that (sin⁡θ−cos⁡θ)(1+sin⁡θcos⁡θ)cos⁡3θ=tan⁡3θ−1.\frac{(\sin\theta-\cos\theta)(1+\sin\theta\cos\theta)}{\cos^{3}\theta}=\tan^{3}\theta-1. Hence, for 0<θ<2π0<\theta<2\pi…
  31. 31Prove the identity sin⁡2A+cos⁡A1−cos⁡2A+sin⁡A=cot⁡A\frac{\sin 2A + \cos A}{1 - \cos 2A + \sin A} = \cot A Hence, solve, for −π≤x≤π-\pi \le x \le \pi, the…
  32. 32In the diagram, OC=4 mOC = 4\text{ m}, OD=7 mOD = 7\text{ m} and angle CBO=angle DOC=angle DAO=90∘CBO = \text{angle } DOC = \text{angle } DAO = 90^\circ.…
  33. 33Do not use a calculator in this question. Given that tan⁡θ=35\tan\theta=\frac{3}{5} and that θ\theta is non-acute, find the e…
  34. 34A particle moves in a straight line such that, tt s after passing through a fixed point OO, its displacement from OO…
  35. 35Given that y=sec⁡xy = \sec x, show that dydx=sec⁡xtan⁡x\frac{dy}{dx} = \sec x \tan x. Hence show that y4−y2=(dydx)2y^4 - y^2 = \left(\frac{dy}{dx}\right)^2…
  36. 36Angles AA and BB are both obtuse and tan⁡(A+B)=34\tan(A + B) = \frac{3}{4}. Without the use of calculators, state the range of a…
  37. 37The diagram shows a triangle PQRPQR. Point MM lies on QRQR such that QM=1QM = 1 unit and MR=3MR = 3 units. Angle PRQ=90∘PRQ = 90^\circ…
  38. 38Given that θ\theta is acute and sin⁡θ=s\sin \theta = s, express, in terms of ss, tan⁡θ\tan \theta sec⁡θ\sec \theta.
  39. 39AA, BB and CC are the angles of a triangle. Given that sin⁡(A+B)=1213\sin(A + B) = \frac{12}{13}. Find the exact value of tan⁡C\tan C…
  40. 40Solve the equation 3cos⁡A+3sec⁡A+7tan⁡A=03\cos A + 3\sec A + 7\tan A = 0 for 0∘<A<360∘0^\circ < A < 360^\circ.
  41. 41Prove that tan⁡x1+sec⁡x+1+sec⁡xtan⁡x=2csc⁡x\dfrac{\tan x}{1+\sec x}+\dfrac{1+\sec x}{\tan x}=2\csc x. Hence, solve tan⁡2x1+sec⁡2x+1+sec⁡2xtan⁡2x=3sec⁡x\dfrac{\tan 2x}{1+\sec 2x}+\dfrac{1+\sec 2x}{\tan 2x}=3\sec x…
  42. 42In the diagram, AC=35 cmAC = 35\text{ cm}, EC=15 cmEC = 15\text{ cm} and angle ACB=θACB = \theta radians. EDED and ABAB are perpendicul…
  43. 43Prove the identity sin⁡x1−cos⁡x+1−cos⁡xsin⁡x=2csc⁡x.\frac{\sin x}{1-\cos x} + \frac{1-\cos x}{\sin x} = 2\csc x. Hence solve the equation sin⁡x1−cos⁡x+1−cos⁡xsin⁡x=5\frac{\sin x}{1-\cos x} + \frac{1-\cos x}{\sin x} = 5…
  44. 44Prove the identity 1+sin⁡xcos⁡x+cos⁡x1+sin⁡x=2sec⁡x\frac{1 + \sin x}{\cos x} + \frac{\cos x}{1 + \sin x} = 2 \sec x. Hence, solve the equation 1+sin⁡xcos⁡x+cos⁡x1+sin⁡x=4\frac{1 + \sin x}{\cos x} + \frac{\cos x}{1 + \sin x} = 4…
  45. 45It is given that cos⁡A=12\cos A = \frac{1}{2} and sin⁡B=−12\sin B = -\frac{1}{\sqrt{2}} where 0∘<A<90∘0^\circ < A < 90^\circ and 180∘<B<270∘180^\circ < B < 270^\circ…
  46. 46A competitive platform diving competition involves one or two divers performing spins, twists and somersaults into a poo…
  47. 47Given that θ\theta is an acute angle and sin⁡θ=m\sin \theta = m, express in terms of mm, Another acute angle α\alpha is su…
  48. 48The function ff is defined by f(x)=3sin⁡x−4cos⁡2xf(x)=3\sin x-4\cos 2x, 0<x<π20<x<\dfrac{\pi}{2}. Explain with working, whether ff is an in…
  49. 49In the diagram, ABAB and CDCD are chords of a circle, with AB=9 cmAB = 9\text{ cm} and CD=25 cmCD = 25\text{ cm}. It is given that…
  50. 50The diagram above shows the side view of a bus stop shelter BCDBCD such that BC=4 mBC = 4\text{ m}, CD=1 mCD = 1\text{ m}, angle…
  51. 51Given that cot⁡θ=−34\cot \theta = -\frac{3}{4} and that tan⁡θ\tan \theta and cos⁡θ\cos \theta have opposite signs, without evaluating…
  52. 52Show that sin⁡105∘=6+24\sin 105^\circ = \dfrac{\sqrt{6} + \sqrt{2}}{4}. The diagram below shows triangle ABCABC, such that AB=4 cmAB = 4\ \text{cm}…
  53. 53A mover needs to deliver a refrigerator to a customer. The refrigerator can be modelled by a cuboid of dimensions 80 cm…
  54. 54It is given that sin⁡A=0.3\sin A = 0.3, where AA is obtuse. Find the following trigonometric ratios. sec⁡A\sec A, cos⁡2A\cos 2A, tan⁡(A+45∘)\tan(A + 45^\circ)…
  55. 55Show that tan⁡x+cos⁡x1+sin⁡x=sec⁡x\tan x + \frac{\cos x}{1 + \sin x} = \sec x. Explain, without the use of calculator, why the equation tan⁡x+cos⁡x1+sin⁡x=−13\tan x + \frac{\cos x}{1 + \sin x} = -\frac{1}{3}…
  56. 56The diagram shows a triangle PQRPQR with an area of 18 cm2^2. The base QRQR has a length of (42+8)(4\sqrt{2} + 8) cm. The li…
  57. 57The diagram above shows two rods ABAB and BCBC of length 4 m and 9 m respectively and ∠ABC=90∘\angle ABC = 90^\circ. Rollers a…
  58. 58Without using a calculator, find the values of the integers aa, bb and cc for which the solution to the equation 5x−3=4+3 x5x - \sqrt{3} = 4 + \sqrt{3}\,x…
  59. 59The diagram shows two right-angled triangles, ABCABC and DECDEC. ∠BAC=∠CDE=θ\angle BAC = \angle CDE = \theta, AB=3.5 cmAB = 3.5\text{ cm}…
  60. 60A curve is such that d2ydx2=−4sin⁡(2x+π)\dfrac{\mathrm{d}^2 y}{\mathrm{d}x^2} = -4\sin(2x+\pi) and the point P(π2,π)P\left(\dfrac{\pi}{2},\pi\right)…