Why not just use a two-way handshake? The short answer is because a two way handshake would only allow one party to establish an ISN(Initial Sequence Number), and the other party to acknowledge it. Which means only one party can send data.
But TCP is a bi-directional communication protocol, which means either end ought to be able to send data reliably. Both parties need to establish an ISN, and both parties need to acknowledge the other's ISN.
Actual use of SYN packet
Sequence number doesn't start at 0. It starts at the ISN (Initial Sequence Number), which is a randomly chosen value. And since TCP is a bi-directional communication, both parties can "speak", and therefore both must randomly generate an ISN as their starting Sequence Number. Which in turn means, both parties need to notify the other party of their starting ISN.
So you end up with this sequence of events for a start of a TCP conversation between Alice and Bob:
Alice ---> Bob SYNchronize with my Initial Sequence Number of X
Alice <--- Bob I received your syn, I ACKnowledge that I am ready for [X+1]
Alice <--- Bob SYNchronize with my Initial Sequence Number of Y
Alice ---> Bob I received your syn, I ACKnowledge that I am ready for [Y+1]
Reference : https://networkengineering.stackexchange.com/questions/24068/why-do-we-need-a-3-way-handshake-why-not-just-2-way