R-formula (a sin + b cos)

Convert a sin x + b cos x = R sin(x + α) where R = √(a²+b²), tan α = b/a.

Y13. Useful for finding max/min and solving equations. The form R sin(x + α) has max R, min −R.

Worked examples

3 sin x + 4 cos x = 5 sin(x + α), tan α = 4/3. Max value 5.

R cos(x − α) form also valid; pick the form the question requests.

Solving 3 sin x + 4 cos x = 2: rewrite as 5 sin(x + α) = 2 ⇒ x + α = arcsin(0.4).

Frequently asked questions

How to choose R sin or R cos?

Question often specifies. Both work; the algebra differs slightly.

α in degrees or radians?

Match the question. A-Level Pure typically radians.