Small-angle approximations and their limits
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| approximation/əˌprɒksɪˈmeɪʃn/ | xấp xỉ |
When is replacing a trigonometric function by the angle itself accurate enough?
- A small turn moves a pointer almost sideways.
- This lesson studies approximation 近似: A nearby simpler value used with a stated accuracy limit.
The method
- When θ is close to zero and measured in radians, sinθ≈θ, tanθ≈θ and cosθ≈1−θ²/2.
- These are approximations, not identities.
- The discarded sine/tangent terms begin at order θ³; the discarded cosine term begins at order θ⁴.
- A smaller absolute angle usually improves these local approximations.
Why does sin θ≈θ fail for an angle entered in degrees?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The small-angle limits come from radian geometry, where arc equals radius times angle. In degrees every value shrinks by π/180, so writing sin 1°≈1 is wrong by a factor of about 57.
A worked case
At θ=0.1 radians, sinθ≈0.1 (actual 0.0998334), tanθ≈0.1 (actual 0.100335) and cosθ≈0.995 (actual 0.995004). Absolute errors are about 0.000167, 0.000335 and 0.00000417. For 6(1−cos x)=0.03, approximate 1−cos x by x²/2: 3x²≈0.03, so x≈±0.1 radians. Check the small-angle assumption after solving; substitution at x=0.1 gives 0.0299750 rather than exactly 0.03. Also sin(2x)/x≈2 for small nonzero x, while 1−cos(2x)≈2x².
Small-angle approximations and their limits
When θ is close to zero and measured in radians, sinθ≈θ, tanθ≈θ and cosθ≈1−θ²/2
Explain each condition before using the corresponding trigonometric formula.
Using the small-angle formula, approximate sin(0.2 radians).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
sinθ≈θ, with θ=0.2 radians.
Where it goes wrong
- Degrees do not work in these formulas: sin(1°)≈π/180, not 1.
- Write ≈ rather than =.
- Subtraction can make relative error more important; preserve the θ² term in 1−cosθ.
- A ratio at θ=0 may be undefined even when its nearby limit exists.
Not true: The formula sinθ≈θ is equally valid for an angle entered in degrees. sinθ≈θ is valid only with θ in radians and small.
Approximate cos(0.2 radians).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
1−0.2²/2=0.98.
The formula sinθ≈θ is equally valid for an angle entered in degrees.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
sinθ≈θ is valid only with θ in radians and small.
Applying it
- Convert to radians first and identify the argument of each trig function: sin(3x)≈3x, not x.
- For a model, compare the approximation error with the precision required.
- If a solution is not small, use the original trigonometric equation; these formulas have no universal accuracy cutoff.
Give the positive approximate solution of 6(1−cos x)=0.03 in radians.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
3x²≈0.03 gives x²≈0.01, so the positive root is 0.1.
Match each small-angle approximation to its first discarded term.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
These are approximations with accuracy limits - write ≈, keep θ in radians, and preserve the θ² term where subtraction would cancel the 1.
In your course
- AQA A-level Mathematics (7357), specification section E.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Approximation: A nearby simpler value used with a stated accuracy limit.
- When θ is close to zero and measured in radians, sinθ≈θ, tanθ≈θ and cosθ≈1−θ²/2.
- Watch for: degrees do not work in these formulas: sin(1°)≈π/180, not 1.