Math topic中文

Solving equations

solution of simple trigonometric equations in a given interval (excluding general solution)

Practice questions

  1. 1Solve the equation 2cot⁡22x=1−5sin⁡2x2 \cot^2 2x = 1 - \frac{5}{\sin 2x} for −180∘≤x≤180∘-180^\circ \le x \le 180^\circ.
  2. 28.8.设△ABC\triangle ABC的内角AA，BB，CC所对的边分别为aa，bb，cc，若bcos⁡C+ccos⁡B=asin⁡Ab\cos C+c\cos B=a\sin A，则△ABC\triangle ABC的形状为…
  3. 3Prove that sin⁡3xsec⁡2x+sin⁡x=tan⁡xsec⁡x\sin^{3} x\sec^{2} x + \sin x = \tan x\sec x. Solve the equation sin⁡3xsec⁡2x+sin⁡x=5\sin^{3} x\sec^{2} x + \sin x = 5 for 0≤x≤360∘0 \le x \le 360^\circ…
  4. 4Given 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}.
  5. 5The equation of a curve is given by y=x+sin⁡2xy = x + \sin^2 x for 0≤x≤π0 \le x \le \pi. Write down the expression for dydx\frac{\mathrm{d}y}{\mathrm{d}x}…
  6. 6Using Rcos⁡(θ−α)R \cos(\theta - \alpha), where R>0R > 0 and 0∘≤α≤90∘0^\circ \le \alpha \le 90^\circ, solve the equation 5cos⁡θ+8sin⁡θ=65 \cos \theta + 8 \sin \theta = 6…
  7. 7The diagram shows a wind turbine with blade 25 m25\text{ m} in length. The centre of the wind turbine is 35 m35\text{ m} fro…
  8. 8Given that the function f(x)=1+2sin⁡23x+4cos⁡23x\text{f}(x) = 1 + 2\sin^2 3x + 4\cos^2 3x, Express f(x)\text{f}(x) in the form a+bcos⁡6xa + b\cos 6x. H…
  9. 9The 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},…
  10. 10Given that sin⁡(A−B)sin⁡(A+B)=23\dfrac{\sin(A-B)}{\sin(A+B)} = \dfrac{2}{3}, prove that tan⁡A=5tan⁡B\tan A = 5\tan B. Hence, or otherwise, solve the…
  11. 11The diagram below shows a circle with chords PQPQ and RSRS intersect at the point XX. P,Q,RP, Q, R and SS are points on t…
  12. 12The 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…
  13. 13A clock is set on the vertical wall on the roof of a building. The distance from the centre of the clock, OO, to the ti…
  14. 14The equation of a curve is y=e−2xtan⁡xy = e^{-2x}\tan x. Show that dydx=e−2x(1−tan⁡x)2\dfrac{dy}{dx} = e^{-2x}(1 - \tan x)^2. Find, in terms of π\pi…
  15. 15Prove that 3cos⁡2x+sin⁡2x=2+cos⁡2x3\cos^2 x + \sin^2 x = 2 + \cos 2x. Hence, using your result from part (a), Hence, using your result from p…
  16. 16The 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}…
  17. 17Prove the identity cos⁡xcosec x−1+cos⁡xcosec x+1=2tan⁡x\frac{\cos x}{\text{cosec } x - 1} + \frac{\cos x}{\text{cosec } x + 1} = 2 \tan x. Solve the equat…
  18. 18A particle moves in a straight line such that its velocity, v m/sv\text{ m/s}, is given by v=9sin⁡(t2)v = 9\sin\left(\frac{t}{2}\right)…
  19. 19The diagram below shows two horizontal lines ABAB and CDCD, where LL is the vertical distance between the lines ABAB an…
  20. 20Prove that csc⁡2x+cot⁡2x=cot⁡x\csc 2x + \cot 2x = \cot x. Hence, deduce the value of cot⁡15∘\cot 15^\circ in surd form. Using part (i), solve…
  21. 21Prove 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…
  22. 22把函数y=sin⁡(2x+4π3)y=\sin (2x+ \dfrac {4π}{3})的图象向右平移φ(φ>0)φ(φ > 0)个单位，所得图象关于yy轴对称，则φφ的最小值为 ___ ．
  23. 23The 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…
  24. 24The diagram shows the front view of a rectangular block ABCEABCE, with dimensions 14 cm by 8 cm, placed on a ramp, VEVE, t…
  25. 25已知函数y=cos⁡xy=\cos x与y=sin⁡(2x+φ)(0⩽φ⩽π)y=\sin (2x+φ)(0\leqslant φ\leqslant π)，它们的图象有一个横坐标为π3 \dfrac {π}{3}的交点，则φ=()φ=() A. π6 \dfrac {π}{6}…
  26. 26The equation of a curve is x2+4y2=20x^2 + 4y^2 = 20 and the equation of a line is y=k−xy = k - x, where kk is a constant. Find th…
  27. 27Prove the identity cos⁡2x−sin⁡2x1+2sin⁡xcos⁡x=1−tan⁡x1+tan⁡x\dfrac{\cos^{2} x - \sin^{2} x}{1 + 2\sin x\cos x} = \dfrac{1 - \tan x}{1 + \tan x}. Solve the equa…
  28. 28已知函数f(x)=2sin⁡ωxcos⁡ωx+23sin⁡2ωx−3(ω>0)f(x)=2\sin ωx\cos ωx+2 \sqrt {3}\sin ^{2}ωx- \sqrt {3}(ω > 0)的最小正周期为ππ． (1)求ωω的值及函数f(x)f(x)的单调减区间； (2)将函数f(x)f(x)…
  29. 29In 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…
  30. 30The 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}…
  31. 3112 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.…
  32. 32The diagram shows an area that is enclosed by glass panels at ABAB, BCBC and ADAD. AB=10 mAB = 10\text{ m}, AD=30 mAD = 30\text{ m}…
  33. 33The diagram shows the side view of a 14 cm by 8 cm rectangular block PQRSPQRS, placed on a ramp, VSVS, tilted at an acute…
  34. 34The diagram shows part of the curve y=2sin⁡2x+1y = 2 \sin 2x + 1. PP is the maximum point of the curve and QQ is the point on t…
  35. 35设△ABC\vartriangle ABC的内角AA、BB、CC所对的边分别为aa，bb，cc，若asin⁡A=bcos⁡C+ccos⁡Ba\sin A=b\cos C+c\cos B，则△ABC\vartriangle ABC的形状为…
  36. 36The expression 8sin⁡θ−3cos⁡θ8 \sin \theta - 3 \cos \theta is defined for 0∘≤θ≤180∘0^\circ \le \theta \le 180^\circ. Using Rsin⁡(θ−α)R \sin(\theta - \alpha)…
  37. 37The height of water in a harbour at time tt hours after midnight on a certain day is modelled by the formula h=4.8sin⁡kπt+5.1h = 4.8 \sin k\pi t + 5.1…
  38. 38The figure shows a piece of cardboard in the shape of a trapezium in which ∠BAD\angle BAD and ∠ABC\angle ABC are right angle…
  39. 39Prove the identity csc⁡4x+cot⁡4x=cot⁡2x\csc 4x + \cot 4x = \cot 2x. Hence, find, for −180∘≤θ≤180∘-180^\circ \le \theta \le 180^\circ, the values of…
  40. 40在△ABC\triangle ABC中，角AA，BB，CC的对边分别是aa，bb，cc，已知a2=b2+c2+bc{{a}^{2}}={{b}^{2}}+{{c}^{2}}+bc，则A=()A=(\quad) A. π3\dfrac{\pi}{3}…
  41. 41a→=(32,sin⁡α)\overrightarrow{a}=(\dfrac{3}{2},\sin \alpha )，b→=(cos⁡α,13)\overrightarrow{b}=(\cos \alpha ,\dfrac{1}{3})且a→/ ⁣/b→\overrightarrow{a}/\!/\overrightarrow{b}…
  42. 42“α=π4α=\dfrac{π}{4}”是“sin⁡α=22\sin α=\dfrac{\sqrt{2}}{2}”的()(\quad) A. 充分不必要条件 B. 必要不充分条件 C. 充要条件 D. 既非充分也非必要条件
  43. 43Prove 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…
  44. 44Calculate, in degrees, the principal values of xx which satisfy 2sin⁡2x+3cos⁡x=02 \sin 2x + 3 \cos x = 0 .
  45. 45The diagram shows the curve y=cos⁡x+sin⁡xy = \cos x + \sin x for 0≤x≤π0 \le x \le \pi. The curve meets the yy-axis at the point AA a…
  46. 46State the principal value of sin⁡−1(−32),thevaluesbetweenwhichtheprincipalvalueof\sin^{-1}\left(-\dfrac{\sqrt{3}}{2}\right), the values between which the principal value of …
  47. 47如果函数y=3cos⁡(2x+φ)y=3\cos (2x+φ)的图象关于点(4π3,0)( \dfrac {4π}{3},0)中心对称，那么∣φ∣|φ|的最小值为()() A. π6 \dfrac {π}{6} B. π4 \dfrac {π}{4}…
  48. 48The diagram shows a rectangular wooden plank ABCDABCD, 5 m by 2 m, leaning against a vertical wall. ADAD makes an acute an…
  49. 49Prove that sin⁡θ(sec⁡θ−tan⁡θ)(sin⁡θ+1)=tan⁡θ\dfrac{\sin \theta}{(\sec \theta - \tan \theta)(\sin \theta + 1)} = \tan \theta. Find all the values of θ\theta…
  50. 50Show 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…
  51. 51The diagram below shows a fenced field PQRSPQRS. It is given that PS=40 mPS = 40\text{ m}, SR=100 mSR = 100\text{ m}, both angles PQRPQR…
  52. 52The point (a,2)(a, 2) is a stationary point on the graph y=x3−x2−x+by = x^3 - x^2 - x + b, where aa and bb are positive constants.…
  53. 53Prove 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…
  54. 54The diagram shows the side view of a 14 cm14\text{ cm} by 8 cm8\text{ cm} rectangular block PQRSPQRS, placed on a ramp, VSVS, t…
  55. 55Factorise sin⁡3x+cos⁡3x\sin^3 x + \cos^3 x completely. Show that sin⁡3x+cos⁡3xsin⁡x+cos⁡x=1−12sin⁡2x\frac{\sin^3 x + \cos^3 x}{\sin x + \cos x} = 1 - \frac{1}{2}\sin 2x…
  56. 56In 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.…
  57. 57A particle moves in a straight line such that, tt s after passing through a fixed point OO, its displacement from OO…
  58. 58已知在△ABC\triangle ABC中，2cos⁡Bsin⁡C=sin⁡A2\cos B\sin C=\sin A，则△ABC\triangle ABC一定为()() A. 等腰三角形 B. 直角三角形 C. 钝角三角形 D. 正三角形
  59. 59A particle PP travels in a straight line and its velocity, vv m/s, is given by v=4sin⁡2(t2)−1v = 4\sin^2\left(\frac{t}{2}\right) - 1…
  60. 60The function ff is defined by f(x)=4cos⁡ax+bf(x) = 4 \cos ax + b for 0≤x≤π0 \le x \le \pi, where aa and bb are constants. The period…