Worked example 1
Convert to wave form
Write 3 cos x + 4 sin x as k cos(x − α), where α is acute
- Compare with k cos x cos α + k sin x sin α.
- Use k cos α = 3 and k sin α = 4.
- Find k = 5 and tan α = .
Answer: 5 cos(x − α), where α = tan−1
Higher Mathematics
Higher Maths Wave functions
Practise wave functions for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Convert a cos x + b sin x to a single sine or cosine form and use it to find ranges and solve equations.
Before you start
- Use addition formulae.
- Solve right-angled triangles.
- Interpret amplitude and range.
Explanation
A linear combination a cos x + b sin x can be written as k cos(x − α) or k sin(x + α). Coefficient comparison finds k and α.
The single-function form makes maximum, minimum and equation solving much clearer.
Visual support
3 cos x + 4 sin x = 5 cos(x − α), where tan α =
Method and rules
- a cos x + b sin x = k cos(x − α).
- k cos α = a and k sin α = b
- k = √(a² + b²), with k > 0
Worked examples
Worked example 2
Method check
What is the maximum value of 3 cos x + 4 sin x?
- The amplitude of the equivalent wave function is k.
So: 5.
Watch out
- Using k = a + b instead of Pythagoras
- Choosing a form whose coefficient signs do not match.
- Finding α but not using the stated interval or quadrant.
Exam reminder
For wave functions, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.