Auxiliary-angle forms, ranges and interval solutions
| English | 中文 | Pinyin |
|---|---|---|
| amplitude/ˈæmplɪtjuːd/ | 振幅 | zhèn fú |
Two periodic effects act together. Can one shifted curve show their combined maximum and minimum?
- Two periodic effects act together. Can one shifted curve show their combined maximum and minimum?
- This lesson studies amplitude 振幅: The nonnegative size of an oscillation about its centre line.
Choose the mathematical structure
- For a and b not both zero, write a cosθ+b sinθ=R cos(θ−α), where R=√(a²+b²)>0, cosα=a/R and sinα=b/R. Choose α in the correct quadrant. The equivalent sine form R sin(θ+β) has sinβ=a/R and cosβ=b/R. A constant c shifts the range to [c−R,c+R]. If a=b=0, the expression is zero and no unique phase is defined.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines amplitude?
The nonnegative size of an oscillation about its centre line.
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.
3cosθ+4sinθ=5cos(θ−α), with cosα=3/5, sinα=4/5 and α≈0.927295 radians. Its sine form is 5sin(θ+β), where β≈0.643501 radians. Thus 2+3cosθ+4sinθ ranges from −3 to 7. For 3cosθ−4sinθ, use 5cos(θ+α). For −3cosθ+4sinθ, the cosine phase is in quadrant II, not the negative principal arctangent. To solve 3cosθ+4sinθ=2 on [0,2π), put δ=arccos(2/5)≈1.159279. Then θ=α±δ+2kπ; the two allowed values are about 2.086575 and 6.051201 radians. A right side of 6 would give no solution, since the amplitude is only 5.
Auxiliary-angle forms, ranges and interval solutions
For a and b not both zero, write a cosθ+b sinθ=R cos(θ−α), where R=√(a²+b²)>0, cosα=a/R and sinα=b/R
Explain how the angle formula follows from the geometric or coefficient conditions.
Find the amplitude of 3cosθ+4sinθ.
R=√(3²+4²)=5.
Test a tempting shortcut
- A plain arctan(b/a) can select the wrong quadrant and fails when a=0. Match both sine and cosine coefficients. The amplitude is √(a²+b²), not |a|+|b|. Do not treat the principal inverse cosine value as the only solution. The angle-mode unit must match the stated interval.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The amplitude of a cosθ+b sinθ is always |a|+|b|. This claim is false. Explain which definition or assumption it violates.
Find the maximum of 2+3cosθ+4sinθ.
The maximum cosine value is 1, so 2+5=7.
The amplitude of a cosθ+b sinθ is always |a|+|b|.
A plain arctan(b/a) can select the wrong quadrant and fails when a=0. Match both sine and cosine coefficients. The amplitude is √(a²+b²), not |a|+|b|. Do not treat the principal inverse cosine value as the only solution. The angle-mode unit must match the stated interval.
Interpret a new situation
- Expand your proposed R cos(θ−α) or R sin(θ+β) to verify every coefficient and sign. For equations, compare the target with the range first, generate both phase branches and translate each by full turns into the original interval. At the upper or lower range endpoint the two branches coincide modulo 2π.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find its minimum.
The minimum cosine value is −1, so 2−5=−3.
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.
The nonnegative size of an oscillation about its centre line. Choose the relationship, show the method, check its assumptions and interpret the result.