finding the period of $\sin(2x+3)$

Question

I tried to find the period of $\sin(2x+3)$;

looking for $p>0$ such that $ \sin(2(x+p)+3)=\sin(2x+3),$ for all $ x \in R$

which means: $ \sin(2(x+p)+3) - \sin(2x+3)= 0 ,$ for all $ x \in R$

that is $\sin(p)\cdot \cos(2x+3+p)=0$, for all $ x \in R$

I'm not sure how to continue....

Answer 1

last equation implies $cos(2x+3+p)=0$ for all $ x\in R$ which is not possible as it is function of $ x$

other solution is $\sin p=0$ for all $x\in R$ which implies $p=n\pi$ where $n \in I$ and $\neq 0$(which would result in $\sin(2x+3)=\sin(2x+3)$

hence smallest period is when $n=1$,ie.$p=\pi$

Answer 2

$$\sin(2x+3)=\sin\left(2\left(x+\frac{3}{2}\right)\right)$$

The period is $$p=\frac{2\pi}{2}=\pi$$