Maths Extension 1 Year 11 - Module 3 - Lesson 9
Auxiliary Angle Method
Name
Date
Class
1. Lesson focus
Auxiliary-angle method. Write a sinx+b cosx as R sin(x+α), where R=√(a²+b²), cosα=a/R and sinα=b/R.
Success criteria
- Apply the lesson method accurately in a routine example.
- Show each algebraic or domain step and check the result.
- Explain why the method and restrictions fit the question.
2. Worked practice
2. State the maximum and minimum of 3sinx+4cosx.
3. Solve 3sinx+4cosx=0 for 0≤x<2π.
3. Challenge
Find the range of 5cosx−12sinx.
4. Multiple choice
The amplitude of a sinx+b cosx is what?
A. √(a²+b²)
B. a+b
C. a²+b²
D. √(a²−b²)
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Write 3sinx+4cosx as R sin(x+α), with 0<α<π/2.
R=5; cosα=3/5 and sinα=4/5, so α=arctan(4/3).
2. State the maximum and minimum of 3sinx+4cosx.
Its amplitude is R=5, so the maximum is 5 and the minimum is −5.
3. Solve 3sinx+4cosx=0 for 0≤x<2π.
Let α=arctan(4/3). 5sin(x+α)=0, so x+α=kπ; solutions x=π−α and 2π−α.
Challenge. Find the range of 5cosx−12sinx.
Amplitude=√(25+144)=13, so the range is [−13,13].
Multiple choice. The amplitude of a sinx+b cosx is what?
Correct answer: A. The coefficients are perpendicular components; the resultant amplitude is their hypotenuse √(a²+b²).
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