Express $2\cos\theta-\sqrt3\sin\theta$ in the form $R\sin(\theta-\alpha)$ , with $R>0$ and $0^\circ<\alpha<360^\circ$ .

Model answer

sqrt7 sin(theta-229.1)

Also accepted: R=sqrt7, alpha=229.1, sqrt(7) sin(theta - 229.1 degrees)

Explanation: Matching coefficients gives

R=\sqrt7

and

\alpha\approx229.1^\circ

(third quadrant, since

R\cos\alpha<0

and

R\sin\alpha<0

), so

2\cos\theta-\sqrt3\sin\theta=\sqrt7\sin(\theta-229.1^\circ)

Derived from ZIMSEC Pure Mathematics Paper 2, November 2025, Q9

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