Auxiliary-angle forms, ranges and interval solutions
Введено| English | Значение |
|---|---|
| amplitude/ˈæmplɪtjuːd/ | амплитуда |
Can one shifted curve show their combined maximum and minimum?
- Two periodic effects act together.
- This lesson studies amplitude 振幅: The nonnegative size of an oscillation about its centre line.
The method
- 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.
Why is the amplitude of 3cos θ+4sin θ equal to 5 rather than 7?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Writing the sum as R cos(θ-α) needs R=√(a²+b²) so that cos and sin of α stay within [-1,1]. |a|+|b|=7 overstates the range; the true range of 2+3cosθ+4sinθ is [-3,7].
A worked case
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.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
R=√(3²+4²)=5.
Where it goes wrong
- 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.
Not true: The amplitude of a cosθ+b sinθ is |a|+|b|. The amplitude is R=√(a²+b²): for 3cosθ+4sinθ it is 5, not 7.
Find the maximum of 2+3cosθ+4sinθ.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The maximum cosine value is 1, so 2+5=7.
The amplitude of a cosθ+b sinθ is |a|+|b|.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
The amplitude is R=√(a²+b²): for 3cosθ+4sinθ it is 5, not 7.
Applying it
- 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.
Find its minimum.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The minimum cosine value is −1, so 2−5=−3.
Match each auxiliary-form quantity to its definition.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
A plain arctan(b/a) can pick the wrong quadrant; matching BOTH coefficients fixes α, and a right-hand side beyond ±R means no solution.
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.
- Amplitude: The nonnegative size of an oscillation about its centre line.
- 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.
- Watch for: A plain arctan(b/a) can select the wrong quadrant and fails when a=0.