Math topic中文

Small angle approximations

Small angle approximations: sin⁡x≈x\sin x \approx x, cos⁡x≈1−12x2\cos x \approx 1 - \tfrac{1}{2}x^2, tan⁡x≈x\tan x \approx x (for xx small, in radians).

Practice questions

  1. 1Given that θ\theta is small, show that sec⁡θ≈1+12θ2\sec\theta \approx 1 + \tfrac{1}{2}\theta^{2}. The diagram below shows a circl…
  2. 2已知α是锐角，且cosα=0.9794，下列各值中，与α最接近的是（ ） A. 73°33′ B. 73°27′ C. 11°37′ D. 11°45′
  3. 3Given that θ\theta is small, show that sec⁡θ≈1+12θ2\sec \theta \approx 1 + \tfrac{1}{2}\theta^2. The diagram below shows a circle…
  4. 4In the triangle ABCABC, AB=1AB = 1, BC=4BC = 4 and angle ABC=θABC = \theta radians. Given that θ\theta is a sufficiently small…
  5. 5In a triangle with vertices A, B and C, angle BACBAC is a right-angle and angle ABC=π3−xABC=\frac{\pi}{3}-x. Show that ABAC=1+3tan⁡x3−tan⁡x\dfrac{AB}{AC}=\dfrac{1+\sqrt{3}\tan x}{\sqrt{3}-\tan x}…
  6. 6A triangle ABCABC is such that AC=2AC = \sqrt{2}, BC=4BC = 4 and angle ACB=14π+θACB = \frac{1}{4}\pi + \theta. Given that heta heta is…
  7. 7The equation xsin⁡x+cos⁡x=1.015x\sin x + \cos x = 1.015 has a positive root α\alpha close to zero. Use small-angle approximations for sin⁡x\sin x…
  8. 8The Environment Authority wishes to build a fence surrounding a park shown by the figure below. BCEF is a square with si…
  9. 9A triangle ABCABC is such that AC=2AC = \sqrt{2}, BC=4BC = 4 and angle ACB=14π+θACB = \tfrac{1}{4}\pi + \theta. Given that θ\theta…
  10. 10In the triangle PQRPQR, angle PQR=π3PQR = \frac{\pi}{3} radians and angle PRQ=π3−θPRQ = \frac{\pi}{3} - \theta radians, where θ\theta…
  11. 11The function h is defined as follows. h:x↦ln⁡(x−k+1), x∈R, x≥k,h : x \mapsto \ln(x - k + 1),\ x \in \mathbb{R},\ x \ge k, where k<0k < 0. Find t…
  12. 12The variables xx and yy are related by (x+y)dydx+ky=2andy=1 at x=0,(x+y)\frac{dy}{dx} + ky = 2\quad\text{and}\quad y = 1\text{ at }x = 0, wher…
  13. 13求极限 lim⁡x→0sin⁡3xx\displaystyle\lim_{x \to 0} \frac{\sin 3x}{x} 的值。
  14. 14Do not use a calculator in answering this question. The complex numbers zz and ww have the same modulus rr, and have…
  15. 15In the triangle ABCABC, angle BAC=θBAC = \theta radians, angle ACB=π6ACB = \frac{\pi}{6} radians and AC=3AC = \sqrt{3}. Given that…
  16. 16In a triangle with vertices AA, BB and CC, angle BACBAC is a right-angle and angle ABC=π3−xABC = \frac{\pi}{3} - x. Show tha…
  17. 17The triangle ABCABC is such that AB=3AB = 3, AC=4AC = 4 and angle BAC=θBAC = \theta radians. The triangle ABCABC is such that AB=3AB = 3…