Small-angle approximations and their limits
| English | 中文 | Pinyin |
|---|---|---|
| approximation/əˌprɒksɪˈmeɪʃn/ | 近似 | jìn sì |
A small turn moves a pointer almost sideways. When is replacing a trigonometric function by the angle itself accurate enough?
- A small turn moves a pointer almost sideways. When is replacing a trigonometric function by the angle itself accurate enough?
- This lesson studies approximation 近似: A nearby simpler value used with a stated accuracy limit.
Choose the mathematical structure
- 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.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines approximation?
A nearby simpler value used with a stated accuracy limit.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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).
sinθ≈θ, with θ=0.2 radians.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The formula sinθ≈θ is equally valid for an angle entered in degrees. This claim is false. Explain which definition or assumption it violates.
Approximate cos(0.2 radians).
1−0.2²/2=0.98.
The formula sinθ≈θ is equally valid for an angle entered in degrees.
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Give the positive approximate solution of 6(1−cos x)=0.03 in radians.
3x²≈0.03 gives x²≈0.01, so the positive root is 0.1.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · E. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A nearby simpler value used with a stated accuracy limit. Choose the relationship, show the method, check its assumptions and interpret the result.