Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 5 / Solution

The addition formula for the sine givesEquating the sine and cosine coefficients gives $A\cos \varphi =\sqrt{3}$ and $A\sin \varphi =1$. Squaring and adding yields $A^2=4$, so $A=2$ because $A>0$. Both coefficients are positive, so $\varphi$ is in the first quadrant. Consequently